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Imagine a waterfall going down the rocks. As it hits each rock, it loses some of its potential energy. Analogously, consider several radiators in a series connected to a boiler that supplies hot water for them. As the water goes from one radiator to another, it cools down, losing its heat energy. The same happens with electrons, as they go along the circuit and meet one resistance after another. Going through each of them, they lose their energy.

Consider the following circuit

Assume, we know how to measure the difference in potential (voltage) between any two points on this circuit. Measuring the voltage between the terminals of the battery gives some value U. Measuring the voltage between the positive terminal of the battery (the longer thin line with a plus sign) and the point in-between the resistors gives some other value U_{1}. Measuring the voltage between the negative terminal of the battery (the shorter thick line with a minus sign) and the point in-between the resistors gives yet other value U_{2}.

Notice that the amount of electricity going through this circuit per unit of time (electric current or amperage) is the same everywhere since it's a closed loop and equals I.

Now let's apply the Ohm's Law to an entire circuit, keeping in mind that the voltage between the terminals of the battery is U and the total resistance of an entire circuit is R=R_{1}+R_{2}: I = U / R = U / (R_{1}+R_{2})

Applying the Ohm's Law to a segment of a circuit that includes only resistor R_{1} and knowing the voltage on its two ends U_{1}, we obtain I = U_{1}/ R_{1}

Equating two different expressions for the electric current I, we can find the voltage U_{1}: I = U / (R_{1}+R_{2}) = U_{1}/ R_{1} from which the value of U_{1} is U_{1} = U·R_{1}/ (R_{1}+R_{2})

Let's do the same calculation for the second resistor. I = U / (R_{1}+R_{2}) = U_{2}/ R_{2} from which the value of U_{2} is U_{2} = U·R_{2}/ (R_{1}+R_{2})

Notice that U_{1} + U_{2} = U. Indeed, U·R_{1}/ (R_{1}+R_{2}) + + U·R_{2}/ (R_{1}+R_{2}) = = U·(R_{1}+R_{2}) / (R_{1}+R_{2}) = U

Completely analogous calculations can be provided with three or more resistors connected in a series. So, the difference in electric potential or voltage drop between the terminals of a battery U is split between the voltage drops on each resistor in a series.

Simply speaking, experiment shows that the current in a conductor (amperage) is proportional to a difference in potentials (voltage) on its ends. This is the Ohm's Law.

Obviously, some physical explanation of this law is appropriate, and this is what this lecture is about.

Electric current is the flow of electrons inside a conductor. Just as a reminder, the traditionally defined direction of the current is opposite to the direction of the flow of electrons.

Even without any electric field electrons inside a conductor are not exactly spin each around its own nucleus, staying on the orbit forever. Some of them, especially those on the outer orbit, can tear off their orbit, jump to another nucleus' orbit, then going somewhere else etc., thus creating certain chaotic environment. When an electric field is present, the chaotic movement of electrons becomes directional to a degree, depending on the strength of the field, thus creating a flow of electrons - an electric current.

This flow of electrons occurs when there is an excess of electrons on one end and deficiency (or less of an excess) of electrons on the other end, which creates an electric field inside a conductor that forces some light electrons to leave their atoms and move, while heavy nuclei with remaining electrons stay put. Electrons are repelled from the end, where there is excess of them and attracted (or repelled less) from the other end.

On their way from one end of a conductor to another these electrons must go through a maze of atoms, many of which have lost some electrons because of the electric field and, therefore, are positively charged, attracting negative electrons in an attempt to compensate lost electrons. Some succeed by capturing free electrons, some not, some lose more electrons in their chaotic but directional movement.

The flow of electrons inside a conductor can be compared with a waterfall from the high level, where potential energy relative to gravitational field of the Earth is greater to a lower level with lesser potential energy.

As water falling down the waterfall, hitting all the stones on its way, electrons hit atoms and lose some of their energy, some of them get absorbed by atoms and don't reach the other end of a conductor, thus diminishing the flow.

The very important aspect in this movement of electrons is the properties of the material a conductor is made of.

Different atoms of different materials have different characteristics of capturing and releasing electrons, when subjected to the forces of an electric field. However, certain common laws of conductivity can be logically explained and used in practical applications.

Ohm's Law

Stronger electric field produced by a greater difference in electric potential (voltage U) between the ends of a conductor should cause more intense movement of electrons, greater number of electrons going from one end of a conductor and reaching the other end per unit of time (amperage I). This qualitative property is perfectly understood. Mathematically speaking, I is a monotonically increasing function of U.

It was an experimental fact that this monotonic function to a very high precision is linear: I = σ·U where σ is called a conductivity of a conductor. In more common cases this law is written not in terms of conductivity, but in terms of resistivity: I = U / R where R is called a resistance of a conductor.

In most cases we will use the Ohm's Law in that last form, using a resistance as the characteristic of a conductor.

Resistance

In practical life we use some conductors with a very low resistance, like copper wire, to connect some electrical appliance, like a regular incandescent lamp bulb, to a source of electric power. In this arrangement we usually consider wiring as having no resistance and concentrate on the properties of an appliance as the one with the electrical resistance R.

Schematically, resistor is pictured as a rectangle or a zigzag line connected by straight lines of wires to a source of electric power

Arrows on this picture show the traditionally defined direction of electricity from positive to negative terminal of the source of electric power - opposite to a direction of electrons' movement.

Consider an extreme case when the resistor is non-conductive, like there is only vacuum in between its two ends. Symbolically, it's equivalent to R=∞. The Ohm's Law in this case results in I=U/R=0, which means that the circuit is broken and there is no current in it.

In an opposite extreme case with R=0 we have I=U/R=∞, which is the so called "short". In practical life it happens when you detach two wires coming to a lamp and connect them. Without the bulb, which has some significant resistance, the electric current in a circuit would almost instantaneously grow very high and you might get electrocuted.

Let's use an incandescent lamp as an example of a resistor. More precisely, the resistor is a tangent spiral (filament) inside the lamp.

Consider what happens with the resistance of this tangent spiral if we will make it longer. Obviously, the electrons will have to go through more obstacles on their way from one end of a spiral to another, which will slow their movement more than when a spiral was shorter. It's logical to expect that doubling the length of a resistor will double its resistance, and it is confirmed by experiments. In general, for linearly-shaped resistors their resistance should be proportional to the length.

The immediate consequence of this consideration is that two resistors, sequentially connected one after another in a series, will have a combined resistance equal to a sum of resistance of each one.

Now, instead of making a spiral longer, let's make it thicker. Electrons will have more space to move, more freedom of direction and more electrons can travel across the spiral per unit of time. Doubling the thickness of a spiral is similar to doubling the width of a highway with more cars moving on it per unit of time. So, the resistance of a thicker spiral will decrease by the factor of increased cross-section area of a tangent spiral. In general, for linearly-shaped resistors their resistance should be inversely proportional to the cross-section area.

Increasing the thickness of a spiral is logically equivalent to using two spirals connected parallel to each other, as on this circuit diagram. The number of electrons per unit of time (electric current or amperage) coming from the common wire is split into two parallel flows, and all the electrons passing through a common part of a circuit per unit of time should be equal to a sum of numbers of electrons passing through each of the parallel segments. The immediate consequence of this consideration is that two resistors, connected parallel to each other, will result in the electric current in a common wire to be a sum of electric currents in each one of them. I = I_{1} + I_{2}

Now let's apply the Ohm's Law to an entire circuit, and each of two parallel branches, keeping in mind that the voltage between the terminals of the battery is U, the same as the voltage between the ends of each resistor, and the resistance of each branch is known as R_{1} and R_{2}.

Assuming the total resistance of two parallel branches is R, the Ohm's Law gives I = U / R

Applying the same principle to one branch with resistor R_{1}, we have I_{1} = U / R_{1}

The same for another branch: I_{2} = U / R_{2}

Now the original equation about a current in the common segment of a wire equaled to a sum of currents in two branches looks like U / R = U / R_{1} + U / R_{2} or 1/R = 1/R_{1} + 1/R_{2}

From the last equation follows the expression for a combined resistance of two resistors installed parallel to each other R = 1 / (1/R_{1} + 1/R_{2}) This formula looks more cumbersome and the one for 1/R above is more commonly occurs.

Instead of resistance, it can be expressed in terms of conductivity of an entire assembly σ=1/R of two parallel resistors with conductivity σ_{1}=1/R_{1} and σ_{2}=1/R_{2}: σ = σ_{1} + σ_{2}

For two identical resistors of resistance r each the combined resistance R would be 1/R = 1/r + 1/r = 2/r R = r/2

More generally, we can derive the resistance of n identical resistors of resistance r each connected parallel to each other. In this case the electric current I in the common wire in split into n identical flows, each having an amperage of I/n. From the Ohm's Law the voltage at the ends of each resistor should be U=I·r/n, from which follows that R=r/n is the resistance of an entire assembly of n identical resistors with resistance r each.

Example

Consider a circuit with resistors R_{1} and R_{2} connected parallel to each other and resistor R_{3} installed after both of them in a series. What would be the resistance R of a combined assembly of these three resistors?

First, determine the resistance of two parallel resistors as a unit. R_{12} = 1 / (1/R_{1} + 1/R_{2}) This this unit of two parallel resistors with resistance R_{12} is in a series with resistor R_{3}, their combined resistance is the sum of their individual resistances R = R_{12} + R_{3} = [1 / (1/R_{1} + 1/R_{2})] + R_{3}

Resistance Units of Measurement

The Ohm's Law allows to easily establish the units of measurement for the resistance of any component in an electric circuit - ohms Ω. From the original form of the Ohm's Law I = U / R follows that R = U / I

This allows to establish a unit of measurement for resistance of any component of an electric circuit. If the difference in potential (voltage) between one end of such a component and another is 1V (volt) and the electric current going through it is 1A (ampere), this component by definition has a resistance of 1Ω (ohm).

Consequently, the resistance of a component in an electric circuit with the current going through it I (ampere) with voltage on its end U (volt) equals to U / I (ohm).

Electric current is a movement of electrons. We know from experience that, when we turn on the switch, the lights in the room are lit practically immediately. Does it mean that electrons from one terminal of a switch go to the light fixture and back to another terminal of a switch that fast? No.

Let's calculate the real speed of electrons, first, theoretically and then in some practical case.

Assume, the amperage of the electric current going through a wire, that is the number of coulombs of electric charge going through a wire per second, is I, and the wire has cross-section area A. Assume further that we know all the physical characteristics of a material our wire is made of, which will be introduced as needed.

Based on this information, our plan is to determine the number of electrons going through the wire per unit of time and, knowing the density of electrons per linear unit of length in the wire, determine the linear speed of these electrons.

Obviously, to determine the linear density of electrons, we will need physical characteristics of a wire.

The number of electrons going through a wire per unit of time is easily determined from the amperageI. Since I represents the number of coulombs of electric charge going through a wire per second, we just have to divide this by the charge of a single electron in coulombse=−1.60217646·10^{−19}C. So, the number of electrons going through a wire per second is N_{e} = I / e.

Now we will determine the linear density of electrons in the wire.

First of all, we have to know how many active electrons in an atom of material our wire is made of participate in the transfer of electric charge, because not all electrons of each atom are freely moving in the electric field, but only those on the outer orbit. Let's assume, this number is n_{e}. Using this number, we convert the number of electrons participating in the transfer of electric charge N_{e} into the number of atoms N_{atoms} in that part of a wire occupied by all electrons transferring the given charge per second I. N_{atoms} = N_{e}/ n_{e} = I / (n_{e}·e).

Next, from the number of atoms we will find their mass and, using the density of wire material, the volume. Dividing the volume by a cross-section of a wire, we will get the length of a segment of wire occupied by those electrons transferring charge per second, which is the speed of electrons or drift.

Knowing the number of atoms, to get to their mass, we will use the Avogadro number N_{A}=6.02214076·10^{23} that represents the number of particles in one mole of a substance. One mole of material our wire is made of is the number of grams equal to its atomic massm_{a}, known for any material used for a wire. If N_{A} atoms have mass of m_{a} gram, N_{atoms} have total mass M_{atoms} = m_{a} · N_{atoms}/ N_{A} = = m_{a}·I / (n_{e}·e·N_{A}).

Knowing the total mass M_{atoms} of all atoms that contain all electrons traveling through a wire per second, we calculate the volume V_{atoms} by using the densityρ of a material the wire is made of. V_{atoms} = M_{atoms}/ ρ = = m_{a}·I / (n_{e}·e·N_{A}·ρ)

Dividing the volume V_{atoms} by the cross-section area of a wire, we will get the length of the wire L occupied by electrons traveling through it in one second L = V_{atoms}/ A = = m_{a}·I / (n_{e}·e·N_{A}·ρ·A) where m_{a} - atomic mass of wire's material (assuming it's one atom molecules, like copper) I - electric current - amperage in the wire n_{e} - number of active electrons in each atom of wire's material that participate in the transfer of electric charge e - electric charge of one electron N_{A} - number of atoms in 1 mol of conducting material - Avogadro Number ρ - density of wire's material A - cross-section area of a wire

The above formula represents the length of a wire occupied by all active electrons traveling through it during one second, which is the speed of movement of electrons making up an electric current, called drift.

Let get to practical examples.

Assume, the voltage or the difference of electric potential E between two ends of a copper wire is maintained at 110V (standard voltage for apartments in the USA). This wire connects a lamp that consumes 120W of electric power P (or wattage). Let the cross-section area of a wire be 3 mm².

First of all, let's calculate the amount of electricity moving through the wire per unit of time - amperage I. As we know, the amperage I, multiplied by voltage E, is the electric power P (wattage). Therefore, E = 110V P = 120W I = P/E = 60W/110V ≅ 1.09A

The atomic number of copper is Z=29. It means, the atom of copper has 29 protons in the nucleus and 29 electrons orbiting a nucleus. These 29 electrons are in four orbits: 2+8+18+1. The outer orbit has only one electron that participates in the movement of electric charge, so the number of active electrons in an atom of copper is n_{e}=1.

The nucleus of an atom of copper has 29 protons and 34 or 36 neutrons, its atomic mass is m_{a} = 63.546g/mol

The electric charge of one electron is e = −1.60217646·10^{−19}C.

The Avogadro number is N_{A}=6.02214076·10^{23}

Density of copper is ρ = 8.96 g/cm³ = 0.00896 g/mm³

Cross-section area of a wire is A = 3 mm²

Using the formula above with values listed, we obtain L = m_{a}·I / (n_{e}·e·N_{A}·ρ·A) we get L ≅ 0.0267 mm/sec This is the speed of electrons traveling along a copper wire in this case. Pretty slow!

Electric current is a flow of electric charge. Since the actual carrier of electric charge is excess or deficiency of electrons, we need certain material where electrons can travel. So, vacuum cannot be a conductor of electricity because there is no electrons in it, but many metals, like copper, can. But we know that the electrons are orbiting the nuclei of the atoms. So, why do they travel?

The answer is: the force of an external electric field pushes or pulls electrons off their orbits and, as a result, they move inside the material where electric field is present towards or away from the source of the electric field, depending on whether the source is positively (deficiency of electrons) or negatively (excess of electrons) charged.

Consider a copper wire. It contains atoms of copper with 29 electrons in each atom, orbiting on different orbits around corresponding nuclei with 29 protons and from 34 to 36 neutrons in each. Electrons stay on their orbit until some outside electric field comes into play. When it does, if its intensity is sufficient to push or pull light electrons off their orbits, while heavy nuclei stay in place, these electrons move in one or another direction as a result of different forces acting on them, the major of which is the intensity of the outside electric field. General direction of electrons is defined by the vector of intensity of the electric field. That makes copper a good conductor of electricity.

On the other hand, there are materials, like glass, where electrons are connected stronger to their nuclei, which makes more difficult to push them off their orbits. these materials do not conduct electricity, they are called insulators or dielectrics.

Ideal conductor, connected to an electrically charged object, makes an extension of this object. Since electrons are freely moving between the original object and an attached conductor, both constitute a new object with an electric charge evenly distributed between its parts. Ideal dielectric, attached to an electrically charged object, does not share its electrons with this object, so the object remains the only one charged.

In practical cases there are no ideal conductors (except under certain conditions of superconductivity under temperatures close to absolute zero) and no ideal dielectrics (except absolute vacuum that has no electrons at all).

Metals are usually good conductors because their nuclei are relatively not easily moved from their places, while electrons are easily pushed off their orbits. We use this property of conductors to direct the electrons to perform some work, like lighting the bulbs or moving electrical cars.

IMPORTANT NOTICE: Conductivity is related to movement of electrons and is a measure of how easily electrons are pushed form their orbits by outside electric field. This should not be mixed with permittivity defined for electric fields and is a measure of propagation of electric field inside some substance.

Electric Current

If the source of the field is a positive charge located near one end of a copper wire, electrons inside the wire would go towards that end. If the negative charge is the source of the field, electrons will move towards the opposite end.

If there is nothing on the opposite end of a copper wire, electrons, after being pushed towards one of the edges, will stop. If, however, there is an opposite charge on the other end of a wire, electrons will move from the negatively charged end to the positively charged one until both charges neutralize each other and whatever end was missing electrons (positively charged) will be compensated by electrons that are in excess on the negatively charged end.

Imagine now that we manage to keep one end of the wire constantly charged positively, while another end constantly charged negatively. Then electrons from the negatively charged end will flow to the positively charged end as long as we can keep these constant opposite charges on both ends. We will have a constant flow of electricity, which is called electric current (or simply current in the context of electricity).

This process of maintaining constant flow of electricity is analogous to maintaining constant flow of water down the water slide using a pump that constantly pumps the water from a pool to the top of a slide, from which it flows down because of the difference in heights and gravity.

While the presence of the electric field is felt almost instantaneously (actually, with a speed close to a speed of light), the electrons that carry electrical charge are not moving from a negatively charged end of a copper wire to the positively charged end with this speed.

A good analogy is the pipe filled with water and a pump connected to one of its ends. As soon as the pump starts working, the water it pumps starts its trip along the pipe and pushes the neighboring molecule of water. Those, in turn, push the next ones etc. So, the water will come from another end of a pipe almost instantaneously (actually, after a time interval needed for the sound waves in the water to cover the length of a pipe), but it's the "old" water already present in the pipe before the pump started working. "New" water that is physically pushed into a pipe by a pump will eventually reach the other end, but not that fast.

Finally, let's talk about measurement of the electric current. The natural way of measurement of the flow of water in the pipe, as exemplified above, would be amount of water flowing out of a pipe per unit of time. In our case of electric charge we can do the same - measure the flow by amount of electricity (in coulombs) traveling from one source of electric field to another (with opposite charge) per unit of time.

The unit of measurement of the electric current is ampere, where 1 ampere is the flow of electricity, when 1 coulomb of electricity is moving across the wire within 1 second. 1 A = 1 C / 1 sec.

Recall the definition of a unit volt as a difference in electric potential between points A and B such that moving one coulomb of electric charge between these points requires one joule of work. Therefore, 1 J = 1 V · 1 C From the definition of ampere above 1 C = 1 A · 1 sec. Therefore, 1 J = 1 V · 1 A · 1 sec 1 V · 1 A = 1 J / 1 sec As we know, 1 J / 1 sec = 1 W (watt) So, electric current of 1 ampere between points with difference of potential 1 volt performs work of 1 watt, that is 1 joule per second.

There is a direct analogy between electricity and mechanics with force analogous to voltage and speed analogous to amperage Force · Distance = Work Force · Distance / Time = = Work / Time = Power Force · (Distance / Time) = = Force · Speed = Power Voltage · Amperage = Power

Let's consider a slightly more complicated example of the electric current. Assume that at one end of a copper wire we have a source of electric field with negative charge and at another end of this wire we have another source of electric field also with negative charge. Both ends will repel electrons inside a wire. However, if the charges are not equal, the larger one will push stronger, and electrons will move away from it towards the other end of a wire.

The situation with two unequal negative charges is analogous to a water pipe with two pumps of different power pumping water into it from both ends. The stronger pump will overcome the weaker and the water will move from a stronger pump to the weaker.

So, the most important factor in determining the direction of electrons in the wire is the intensity of electric field produced by electric charges. For multiple sources of electric field their vectors of intensity are added. From a general viewpoint, if there is a difference in intensity of electric fields, electrons will travel in the direction defined by a stronger force. In practical situation, when two sources of electricity are applied to two ends of a wire, one positive and one negative, one end attracts electrons and another pushes them away, the flow of electrons will be always from negative to positive charge.

Assume, the intensity of electric field at the end A of a wire is E_{A} and intensity at the other end B is E_{B}. If both charges at points A and B are positive or both negative, the vectors E_{A} and E_{B} inside a wire are oppositely directed. If the charges are of different sings (which is a typical situation in practical applications of electricity), these vectors are directed the same way.

The force acting on each coulomb of electricity inside a wire is a vector sum of both intensities: E = E_{A} + E_{B} The work needed to move one coulomb of electricity is, therefore, W = E·L, where L is the length of a wire. This value W represents the difference of electric potentials of the electric field between points A and B, that is the voltageV_{AB} between them. The difference in intensity of an electric field corresponds to the non-zero voltage between these points.

If we can maintain the difference in electric field's potential between the two ends of a wire (non-zero voltage between them), the intensity of an electric field will push electrons from one end of a wire to another. This is how direct electric current is maintained.

As electrons move from one end to another, they leave "holes" - spots where they used to be, which are "moving" in the opposite direction. Since we conditionally associate "negative" charge with electrons and "positive" charge with the absence of electrons ("holes"), we can say that the direction of positive charges is opposite to that of negative.

For historical reasons, because electrons were not discovered yet, the direction of positive charges (that is, "holes" that are left, when electrons leave their places) was defined as a direction of the electric current.

The word direct means that the direction of the flow of electrons does not change with time and goes from the end with negative charge to the end with positive charge, which implies that the direction of the electric current (the direction of "holes" left by electrons) is opposite, from positive to negative end. For practical reasons we will not consider the case of the same sign of charges on both ends.

In most practical cases there is a device that separates the electrons from the neutral atoms within some object, thereby producing negative and positive charges on its terminals. If there is some conductor of electricity between these terminals, electrons will move from one terminal to another along this conductor, which constitutes a direct electric current in it.

Please refer to Problems 4 of "Electric Field", as we will use its results. The problem there derives a formula of intensity of the electric field produced by an infinitely thin disk of radius R charged with surface density of electricity σ at point P positioned at height h above the center of this disk in case the space around this disk is filled with material with dielectric constant (also known as relative permittivity) ε_{r}.

The direction of the vector of electric field intensity is along the perpendicular from point P to a charged disk and its magnitude equals to E(h) = = [σ/(2ε_{r}·ε_{0})]·[1−1/√1+R²/h² ]

From the above formula for any media filling the space around a charged disk we see that the greater dielectric constantε_{r} for the media that fills the space around the charged disk - the smaller intensity of the field around it.

Assume now that our goal is to generate electric charge (excess or deficiency of electrons) and store it somehow. Since electrons are not easily produced from nothing, we should take some electrically neutral object X, separate part of its electrons from the atoms and place them into a different object X_{−}, which becomes negatively charged, leaving old object positively charged, which we can label now X_{+}.

For practical reasons objects X_{−} and X_{+} should be near each other and we should prevent any kind of exchange of electric charge, like a spark, between them, which would negate our efforts to separate negative and positive charges. Considering the proximity of X_{−} and X_{+}, it would be beneficial for avoiding any exchange of the charge between them to have a uniform distribution of charge in each object.

The best configuration of objects X_{−} and X_{+} that allows approximately uniform distribution of charge in the presence of opposite charge nearby is when both objects are thin flat plates positioned parallel to each other. Other configurations will cause the concentration of charges in places close to the opposite charge and higher intensity of the electric field between them, which might result in a discharging spark. Another configuration might be of two concentric spheres, but it's not very practical.

Consider now two relatively large infinitely thin disks of radius R with opposite charges +Q and −Q positioned parallel to each other, perpendicular to a line connecting their centers and at distance d from each other. Let's measure an intensity of the electric field at any point between these disks on a center line at distance h from positively charged disk (0 ≤ h ≤ d).

The intensity of a combined field of two disks is a vector sum of intensities of all components of this field. The direction of the intensity of both fields is along the center line between them. The probe charge is +1C, so the positively charged disk will repel it, while negatively charged one will attract it. So, we can calculate the magnitude of each field and add them together to get the magnitude of the combined field.

The density of the electric charge for these disks equals to a total charge Q divided by a surface area A. There are two opposite surfaces of each disk, but in case we have two close to each other parallel disks with opposite charges the electric charge of each disk (excess or deficiency of electrons) concentrates on a surface that is close to another disk with an opposite charge. So for each disk the absolute value of charge density is constant that equals to σ = Q/A = Q/(πR²)

The magnitude of the field intensity from a positively charged disk at distance h from its surface is E(h) = = [σ/(2ε_{r}·ε_{0})]·[1−1/√1+R²/h² ]

The magnitude of the field intensity from a negatively charged disk at distance d−h from its surface is E(d−h).

Just as an observation, let's notice that, in case the disks are very large and the distance between them very small, both E(h) and E(d−h) are approximately equal to E = σ/(2ε_{r}·ε_{0})

Since both vectors of intensity are directed from positive to negative disk (positively charged disk repels the probe charge of +1C positioned in-between the disks, and the negatively charged disk attracts it), the magnitude of the intensity of the combined electric field produced by both disks equals to E(h) = E(h)+E(d−h) Expressed in all its details, the formula is quite large and difficult to analyze.

At this point we will do what physicists usually do with a cumbersome formula - assume that in practice certain really small values can be assumed as infinitely small and certain large values to be infinitely large. Indeed, if h→0 or h→d the assumption above is correct. Representing graphically intensity E as a function of h for R=10 and d=0.1 on a segment 0 ≤ h ≤ d shows hardly visible rise in the middle of a segment with h=d/2.

So, for practical reasons we will assume that the distance between the disks d is very small, while the radius of disks R is very large. Since the point we measure the intensity is between the disks at the distance h from one of them, variables h and d−h can also be assumed as very small. This assumption leads to consider the values R²/h² and R²/(d−h)² as infinitely large, which result in the following approximate formula for the intensity of a combined field between the disks: E ≅ σ/(2ε_{r}·ε_{0}) + σ/(2ε_{r}·ε_{0}) = = σ/(ε_{r}·ε_{0}) So, approximately, the electric field between the large parallel disks on a small distance from each other is uniform and depends only on the density of electric charge on the disks σ and the dielectric constant of the media between them ε_{r}.

Let's analyze a process of discharging of electricity between two disks. Since our purpose is to store the charges and to avoid discharge, we should know how much electricity Q we can store in these two disks before electrons jump from a negatively charged disk to a positively charged because of the force of attraction.

The discharge will be more difficult if the force of attraction between the disks is less. The force of attraction is characterized by the intensity of the field between the disks. Knowing intensity of the field between the disks E, which is the force acting on a unit charge, we can calculate the work needed by a charge of +1C to overcome a distance d between the disks by multiplying the intensity (force) by the distance.

The work needed to discharge +1C, that is the work needed to move +1C of charge from one disk to another (the voltage between the disks) is V = E·d = σ·d/(ε_{r}·ε_{0})

Let's recall that the purpose of our work is to store as much electric charge in these two disks as possible. But with growing charge Q proportionally grows the density of electricity σ and proportionally grows the intensity of the field and the voltage between the disks.

Let's introduce the new characteristic that defines the ability of our two disk construction, called capacitor, to hold electric charge - capacityC = Q/V. Defined as such, the capacity of a capacitor described above equals to C = Q/V = σ·A/[σ·d/(ε_{r}·ε_{0})] = = A·(ε_{r}·ε_{0})/d

As we see, the capacity of our capacitor depends on three major factors: (a) the area of the disks (capacity is proportional to this area) (b) the distance between disks (capacity is inversely proportional to this area) (c) the dielectric constant of the media between the disks (capacity is proportional to a dielectric constant of the media).

To satisfy the purpose of storing as much electricity in the capacitor as possible, we should increase the area of the disks, decrease the distance between the disks and isolate disks from each other with the media with high dielectric constant.

The disk shape of a capacitor is not really important, as long as it contains two flat parallel shapes on a small distance from each other. The easiest variation from the practical standpoint is two rectangles. Although different shape leads to slightly different results in the value of capacity, the difference is negligible, and it disappears completely when the distance between planes becomes an infinitesimal value. So, the definition of capacity C = A·(ε_{r}·ε_{0})/d is quite universal and is widely used as a characteristic of a capacitor.

Finally, if we adapt a rectangular form of a capacitor, it's easy to significantly increase the area A by folding or rolling both surfaces together. This technique is actually utilized in manufacturing capacitors of large capacity and it's called parallel connection of capacitors. In such a way of connecting capacitors the total capacity of an assembly equals to a sum of capacities of individual capacitors C = C_{1}+C_{2}+C_{3}+C_{4}+... This picture depicts an idea to stack smaller individual capacitors to enlarge the area, thus allowing to store more electric charge in a smaller volume.

Recall the concept of the electric field potential. By definition, the electric field potential is a quantitative characteristic of an electric field, defined for each position in this field, as the amount of work needed to move a probe - positively electrically charged point-object of +1C (one coulomb) - from infinitely remote point in space, where the field does not exist, to this position in the field.

Also recall that in an electric field amount of work to move a charge from one position to another is independent of a trajectory because electrostatic forces are conservative. So, amount of work to move a charge from point A to point B along a straight line between them is the same as if we move along some curve or go from A to an infinitely far point and then return to B.

Electric potential for each point of an electric field fully defines this field. If we know the electric potential at each point of a field, we don't have to know what kind of an object is the source of the field, nor its charge, nor shape in order to understand the movement of any probe object in this field.

To find the amount of work needed to move a charge q from a point in the electric field with a potential V_{1} to a point with potential V_{2} we can simply multiply the electric charge by the difference of electric potentials between these points: W = q·(V_{2}−V_{1}) The sign of the resulting value of work W is important. It signifies whether outside force has to perform the work against the forces of the electric field (like forcing a positive charge to go further from the attracting negative charge) or the electric field does the work itself (like forcing a positive charge to go further from the repelling positive charge).

The expression in the parenthesis in the above formula for work W, that signifies the difference in electric potentials between two points in the electric field, is the main component to calculate the work needed to move a charge between these points. This expression has a special name - voltage - in honor of Italian physicist Alessandro Volta.

Since the electric potential is the work performed on the unit charge, the convenient unit of measurement of this potential is the unit of work per unit of charge.

This unit of measurement is called volt (symbol V) and it is defined as such a difference in electric potential between two points in an electric field that one joule of work (1J) is required to move one coulomb of positive charge (+1C) between these points: 1V = 1J/1C

In the electric field produced by an electrically charged point-object with Q amount of electricity the potential at any point depends only on its distance R from the source of the field and equals to k·Q/R. So, the voltage between two points at distances R_{1} and R_{2} equals to ΔV = k·Q·(1/R_{1} − 1/R_{2})

In the electric field produced by an electrically charged infinite plane with density of charge σ the intensity of the field is constant for all points in space and is equal to 2π·k·σ (see Problems 2 of this section). The direction of the force is always perpendicular to the plane. Therefore, moving a probe charge parallel to the plane does not require any work, and the only parameter we have to take into consideration is the height above the plane. The amount of work needed to move a probe charge of +1C from the height H_{1} to height H_{2} is the product of the force (intensity of the field) by distance (difference in height). Therefore, it equals to ΔV = 2π·k·σ·(H_{2}−H_{1})

Important Analogy

Producing electric charge by separation of electrons from the neutral atoms can be compared with raising the level of water (or any other liquid) in the tall vertical tube with closed valve at the bottom in the gravitational field of Earth. Electric charge causes the existence of electric field that is the source of a force on any electrically charged object. The water raised to a certain height is the source of pressure, which is the force acting on any object at the bottom. The voltage, which is a difference in electric potential between two points in the electric field is analogous to a difference in potential energy between the water at two different heights. Connecting positive and negative charges will cause their mutual neutralization, so no electric field and no voltage will be present anymore. Opening a valve in the vertical tube with water will cause the water to flow down, and no pressure will exist anymore at the bottom of the tube.

This analogy might be useful to understand many facts related to electricity, and it goes much deeper than just about voltage. Electric generators, electric motors, conductors, resistors etc. - all can be to an extent compared with corresponding water-based devices. We will address these concepts, as we progress with the course.

An infinitesimally thin disk α of radius R is electrically charged with uniform density of electric charge σ (coulombs per square meter). What is the intensity of the electric field produced by this disk at point P positioned at distance h (meters) from its surface on a perpendicular through its center?

Solution Please refer to Problems 2 of "Electric Field", as we will use its results. The Problem A from Problems 2 is about intensity of the electric field produced by an infinite plane.

Here we will consider a finite electrically charged infinitely thin disk of the radius R instead of an infinite plane as in Problems 2. Assume, as in Problems 2, the density of electrical charge of this disk is σ and the point we want to measure the intensity of the field is P located above the disk on the perpendicular through its center on height h above it.

Going through exactly the same logic as in Problems 2, dividing our disk into concentric rings, we will come to a formula for intensity produced by an infinitely narrow ring of inner radius r and outer radius r+dr, where r is changing from 0 to R: dE(h,r) = = 2π·k·σ·r·dr·h / (h²+r²)^{3/2}

After substitution y = 1 + r²/h² we obtain dy = 2r·dr/h² 2r·dr = h²·dy dE(h,y) = = π·k·σ·dy·h³ / (h²+r²)^{3/2}= = π·k·σ·dy / (1+r²/h²)^{3/2} = = π·k·σ·y^{−3/2}·dy which is easy to integrate since it's a plain power function. The limits of integration for r are from 0 to R. Therefore, the limits of integration for y are from 1 to 1+R²/h². The indefinite integral of y^{−3/2} is −2y^{−1/2}. Therefore, the total vector of intensity at point P equals to E(h) = = −2π·k·σ·[(1+R²/h²)^{−1/2}−1] = = 2π·k·σ·[1−(1+R²/h²)^{−1/2}] = = 2π·k·σ·[1−1/√1+R²/h² ]

We can rewrite this formula using the permittivity of vacuum ε_{0}=1/(4π·k) as E(h) = (σ/2ε_{0})·[1−1/√1+R²/h² ]

Analyzing this formula, we see that the intensity of the electric field of a uniformly charged disk of radius R at point above its center on the height h depends on the ratio R/h. If the height remains the same, but the radius increases to infinity, the formula transforms into the one we obtained in the Problem 2 for infinite charged plane. If the height h decreases to zero with a fixed radius R, the intensity gradually increases to its maximum value 2π·k·σ, which is the same as for an infinite plane in Problems 2. So, for a small height the uniformly charged disk acts like an infinite plane. If the height h increases to infinity with a fixed radius R, the intensity gradually decreases to zero. All conclusions are intuitively obvious.

Another parameter from which the intensity depends is the medium around a charged object. Knowing from a previous lecture about permittivity, we can consider the space around the charged disk to be not only vacuum, but any media with known dialectic constantε_{r}. In this case, instead of Coulomb's constant k, we have to use 1/(4π·ε_{r}·ε_{0}) and the formula for intensity looks like this: E(h) = = [σ/(2ε_{r}·ε_{0})]·[1−1/√1+R²/h² ]

From the above formula for any media filling the space around a charged disk we see that the greater dielectric constant for this media - the smaller intensity of the field around it.

Let's take a closer look at Coulomb's Law F = k·q_{A}·q_{B}/ R² where F is the magnitude of the force of attraction (in case of opposite charges) or repelling (in case of the same type of charge, positive or negative) in N - newtons q_{A} is electric charge of point-object A in C - coulombs q_{B} is electric charge of point-object B in C - coulombs R is the distance between charged objects in m - meters k is a coefficient of proportionality (Coulomb's constant) equals to 9.0·10^{9} in N·m²/C²

The intuitive explanation of the inverse proportionality of this force to a square of a distance between objects A and B was that the force emitting by an electric point-charge is distributed around it in a radial fashion and, at distance R, should be inversely proportional to an area of a sphere of the radius R. The area of a sphere of radius R is 4πR². Therefore, it's more natural to express Coulomb's with 4πR² in the denominator. Then it will look like this F = q_{A}·q_{B}/ (4π·ε_{0}·R²) where ε_{0} is a constant called permittivity of vacuum. In terms of Coulomb's constant k it is equal to ε_{0} = 1/(4π·k) = 8.85419·10^{−12} measured in C²/(N·m²)

In the above definition permittivity is the property of space between the charges to let the force of electric field through. It is analogous to such mechanical properties as resistance, friction, viscosity.

As experiments show, the same electric charges at the same distance but in different environments produce electric fields of different intensities. Environment matters. In vacuum a specific point-charge at a specific distance produces the field of one intensity, while, positioned inside a sand box, the same charge at the same distance would produce a field of different intensity.

That's why we specifically called ε_{0} the permittivity of vacuum, as no other environment was considered. All the experiments described before relate to vacuum as the media where these experiments are conducted. In different environment the force of electric field would differ.

This prompts us to introduce an absolute permittivityε_{a} (or simply ε) of any media and its relative permittivityε_{r}=ε_{a}/ε_{0}. The relative permittivity of a media is also called its dielectric constant with the value of this dielectric constant for vacuum being equal to 1. For olive oil the dielectric constant is 3, for silicon its about 11-12, for mineral oil its about 2, for marble - 8, for titanium dioxide - between 86 and 173 etc.

Now the formula for intensity of electric field in a media with relative permittivityε_{r} looks like this F = q_{A}·q_{B}/ (4π·ε_{r}·ε_{0}·R²) So, generally speaking, when the electric field speads into any media with a dielectric constantε_{r}, we should use the coefficient 1 /(4π·ε_{r}·ε_{0}) instead of Coulomb's constant k.

The greater the value of the dielectric constant - the stronger it resists to penetration of electric field, so the field is weaker than in vacuum for the same charge and distance. Vacuum is the easiest for the electric field to penetrate.

Also worth noting that the permittivity of any material depends on its temperature and exact chemical composition. This allows, for example, to measure the temperature or humidity of air by measuring its relative permittivity.

Materials with high value of permittivity are used for electrical insulation to prevent the electric field from dissipating around electrical charges.

An infinitesimally thin sphere α of radius R is electrically charged with uniform density of electric charge σ (coulombs per square meter). A point P is inside this sphere at a distance h (meters) from its center, so h is less than R. What is the intensity of the electric field produced by this sphere at point P?

Solution

The magnitude of the field intensity at point P produced by any infinitesimal piece of sphere α is inversely proportional to a square of its distance to point P and directly proportional to its charge, and the charge, in turn, is proportional to its area with density σ being a coefficient of proportionality.

Let's define a system of Cartesian coordinates in our three-dimensional space with the origin at the center O of the center of the charged sphere α with Z-axis along segment OP.

Then the coordinates of point P, where we have to calculate the vector of electric field intensity, are (0,0,h).

From the considerations of symmetry, the vector of intensity of the field, produced by an entire electrically charged sphere α, at point P inside it should be directed along a line OP from point P to a center of a sphere, that is along Z-axis. Indeed, for any infinitesimal area of sphere α near point (x,y,z) there is an area symmetrical to it relatively to Z-axis near point (−x,−y,z), which produces the intensity vector of the same magnitude, the same vertical (parallel to OP) component of it and opposite horizontal component. So, all horizontal components will cancel each other, while vertical ones can be summarized by magnitude. Therefore, we should take into account only projections of all individual intensity vectors from all areas of a sphere onto Z-axis, as all other components will cancel each other.

The approach we will choose is to take an infinitesimal area on a sphere in a form of a spherical ring of infinitesimal width, produced by cutting a sphere by two planes parallel to XY-plane at Z-coordinates z=r and z=r+dr and calculate the vertical component of the intensity vector produced by it. Then we will integrate the result from r=−R to r=R. This choice is based on a simple fact that for every small piece of this spherical ring its distance to point P is the same, as well as an angle between its vector of intensity and Z-axis is the same, hence the vertical component of the field intensity vector produced by it will be the same as for any other such piece of this spherical ring, if it has the same area, while the horizontal component of the intensity vector will be canceled by a symmetrical piece of this spherical ring lying diametrically across it.

Since any infinitesimal part of this ring has exactly the same vertical component of the intensity vector as any other part having the same area, to get the total vertical component of intensity for an entire ring, we can use its total charge that depends on its total area and charge density σ.

The area of a spherical ring equals to a difference between areas of two spherical caps. The area of a spherical cap equals to 2π·R·H, where H is the height of a cap. The spherical cap formed by a plane cutting a sphere at z=r has height R−r. The spherical cap formed by a plane cutting a sphere at z=r+dr has height R−r−dr. Therefore, the area of a spherical ring between these two cutting planes is area(r,dr) = = 2π·R·[(R−r)−(R−r−dr)] = = 2π·R·dr

The charge concentrated in this spherical ring is dQ(r) = 2π·σ·R·dr

To find the magnitude of the intensity vector produced by this ring we need to know its charge (calculated above) and the distance to a point, where the intensity is supposed to be calculated. This distance can be calculated using the Pythagorean Theorem: L² = (R² − r²) + (h−r)² = = R² + h² −2h·r The magnitude of the intensity vector produced by this spherical ring is, therefore, dE(h,r) = k·dQ(r)/L² = = 2π·k·σ·R·dr / L² = = 2π·k·σ·R·dr / (R²+h²−2h·r)

We are interested only in vertical component of this vector, which is equal to dE_{z}(h,r)= dE(h,r)·sin(∠PAB) = = dE(h,r)·(h−r)/L = = 2π·k·σ·R·dr·(h−r) / L³ = = 2π·k·σ·R·dr·(h−r) / (R²+h²−2h·r)^{3/2}

Integrating this by r from −R to R can be done as follows. First of all, let's substitute x = R²+h²−2h·r Then r = (R²+h²−x)/2h dr = −dx/2h The limits of integration for x are from R²+h²+2h·R=(R+h)² to R²+h²−2h·R=(R−h)². Now the expression to integrate looks like dE_{z}(h,x) = C·(h²−R²+x)·x^{−3/2}·dx where constant C equals to C = −π·k·σ·R/(2h²)

Let's integrate the above expression in the limits specified. First, find the indefinite integral. ∫C·(h²−R²+x)·x^{−3/2}·dx = = −2C·(h²−R²)·x^{−1/2} + 2C·x^{1/2} = = 2C·[(R²−h²)·x^{−1/2} + x^{1/2}] This expression for indefinite integral should be used to calculate the definite integral in limits for x from (R+h)² to (R−h)².

Assuming that point P is inside a sphere, that is h is less than radius R, ((R+h)²)^{1/2} = R+h ((R−h)²)^{1/2} = R−h

Therefore, substituting the upper limit into an expression for an indefinite integral, we get 2C·[(R²−h²)/(R−h) + (R−h)] = 4C·R Substituting the lower limit, we get 2C·[(R²−h²)/(R+h) + (R+h)] = 4C·R The difference between these two expressions is zero, which means that the intensity of the electric field inside a uniformly charged sphere is zero.

Problem B

An infinitesimally thin sphere α of radius R is electrically charged with uniform density of electric charge σ (coulombs per square meter). A point P is outside this sphere at a distance h (meters) from its center, so h is greater than R. What is the intensity of the electric field produced by this sphere at point P?

Solution

Start as in the previous problem up to indefinite integral 2C·[(R²−h²)·x^{−1/2} + x^{1/2}]

Assuming that point P is outside a sphere, that is h is greater than radius R, ((R+h)²)^{1/2} = R+h ((R−h)²)^{1/2} = h−R

Therefore, substituting the upper limit into an expression for an indefinite integral, we get 2C·[(R²−h²)/(h−R) + (h−R)] = −4C·R Substituting the lower limit, we get 2C·[(R²−h²)/(R+h) + (R+h)] = 4C·R The difference between them is the intensity of the electric field produced by a sphere at a point outside it: E(h) = −8C·R = = 8π·k·σ·R²/(2h²) = = 4π·k·σ·R²/h²

Notice that area(Sphere) = 4π·R² Therefore, Q = 4π·σ·R² where σ is the density of electric charge on a sphere, represents a total charge of a sphere. Hence, 4π·k·σ·R²/h² = k·Q/h² and the field intensity of a sphere at a point outside it equals to intensity of a point-object with the same electric charge and located at the center of a sphere.

An infinite infinitesimally thin plane α is electrically charged with uniform density of electric charge σ (coulombs per square meter). What is the intensity of the electric field produced by this plane at point P positioned at a distance h (meters) from its surface?

Solution

The magnitude of the field intensity at point P produced by any infinitesimal piece of plane α is inversely proportional to a square of its distance to point P and directly proportional to its charge, and the charge, in turn, is proportional to its area with density σ being a coefficient of proportionality.

Let's define a system of cylindrical coordinates in our three-dimensional space with the origin at the projection O of the point P onto our electrically charged plane α with Z-axis along segment OP and polar coordinates on plane α with r for radial distance OA between any point A on this plane and the origin of coordinates O and φ for a counterclockwise angle from the positive direction of some arbitrarily chosen base ray OX within plane α, originated an point O, to radius OA.

Then the coordinates of point P, where we have to calculate the vector of electric field intensity, are (0,0,h). The coordinates of any point A on plane α will be (r,φ,0).

From the considerations of symmetry, the vector of intensity of the field, produced by an entire infinite electrically charged plane α, at point P outside of it should be directed along a perpendicular OP from point P to a plane. Indeed, for any infinitesimal area of plane α there is an area symmetrical to it relatively to point O, which produces the intensity vector of the same magnitude, the same vertical (parallel to OP) component of it and opposite horizontal (within a plane α) component. So, all horizontal components will cancel each other, while vertical ones can be summarized by magnitude. Therefore, we should take into account only projections of all individual intensity vectors from all areas of a plane onto a perpendicular OP from point P onto plane α, as all other components will cancel each other.

The approach we will choose is to take an infinitesimal area on a plane in a form of a ring centered at point O of infinitesimal width dr with inner radius r and outer radius r+dr and calculate the vertical component of the intensity vector produced by it. Then we will integrate the result from r=0 to infinity. This choice is based on a simple fact that for every small piece of this ring its distance to point P is the same, as well as an angle between its vector of intensity and Z-axis is the same, hence the vertical component of the field intensity vector produced by it will be the same as for any other such piece of this ring, if it has the same area, while the horizontal component of the intensity vector will be canceled by a symmetrical piece of this ring lying diametrically across it.

Since any infinitesimal part of this ring has exactly the same vertical component of the intensity vector as any other part having the same area, to get the total vertical component of intensity for an entire ring, we can use its total charge that depends on its total area and charge density σ.

The area of a ring equals to area(r,dr) = π[(r+dr)²−r²] = = 2π·r·dr + π·(dr)² We can drop the infinitesimal of the second order π·(dr)² and leave only the first component - infinitesimal of the first order, that we plan to integrate by r from 0 to infinity.

The charge concentrated in this ring is dQ(r) = 2π·σ·r·dr

To find the magnitude of the intensity vector produced by this ring we need to know its charge (calculated above) and the distance to a point, where the intensity is supposed to be calculated. This distance can be calculated using the Pythagorean Theorem: L² = h² + r² The magnitude of the intensity vector produced by this ring is, therefore, dE(h,r) = k·dQ(r)/L² = = 2π·k·σ·r·dr / L² = = 2π·k·σ·r·dr / (h² + r²)

We are interested only in vertical component of this vector, which is equal to dE_{z}(h,r)= dE(h,r)·sin(∠PAO) = = dE(h,r)·h/L = = 2π·k·σ·r·dr·h / L³ = = 2π·k·σ·r·dr·h / (h² + r²)^{3/2}

Integrating this by r from 0 to ∞ can be done as follows. First of all, let's substitute x = r/h Then r = h·x dr = h·dx The limits of integration for x are the same, from 0 to ∞. Now the expression to integrate looks like dE_{z}(h,x) = = 2π·k·σ·h³·x·dx / h³(1 + x²)^{3/2} = = 2π·k·σ·x·dx / (1 + x²)^{3/2} Before going into details of integration, note that this expression does not depend on distance h from point P to an electrically charged plane α. This is quite remarkable! No matter how far point P is from plane α, the intensity of electric field at this point is the same.

To integrate the last expression for a projection onto Z-axis of the intensity of electric field produced by an infinitesimal area of plane α, introduce another substitution: y = x² + 1 Then x·dx = dy/2 The limits of integration for y are from 1 to ∞. The expression to integrate becomes dE_{z}(y) = π·k·σ·dy / y^{3/2} = = π·k·σ·y^{−3/2}·dy

This is easy to integrate. The indefinite integral of y^{n} is y^{n+1}/(n+1). Using this for n=−3/2, we get an indefinite integral of our function −2·π·k·σ·y^{−1/2} + C Using the Newton-Leibniz formula for limits from 1 to ∞, this gives the value of the magnitude of the total intensity of a charged plane α: E = ∫_{[1,∞]}π·k·σ·y^{−3/2}·dy = 2π·k·σ

Let's note again that this value is independent of the distance h of point P, where we measure the intensity of the electric field, from an electrically charged plane α. It only depends on the density σ of electric charge on this plane.

As for direction of the intensity vector, as we suggested above, it's always perpendicular to the plane α. Hence, we can say that the electric field produced by a uniformly charged plane is uniform, at each point in space it is directed along a perpendicular to a plane and has a magnitude E=2π·k·σ, where σ represents the density of electric charge on a plane and k is a Coulomb's constant.

Problem B

An infinite infinitesimally thin plane α is electrically charged with uniform density of electric charge σ (coulombs per square meter). What is the work needed to move a charge q (coulombs) from point M positioned at a distance m (meters) from its surface to point N positioned at a distance n (meters) from its surface?

Solution

Notice that positions of points M and N are given only in terms of their distance to a charged plane α, that is in terms of vertical displacement. Distance between them in the horizontal direction is irrelevant since any horizontal movement will be perpendicular to the vectors of field intensity and, therefore, require no work to be done.

So, our work only depends on the distance along the vertical and can be calculated as W_{MN} = E·(n−m)·q = = 2π·k·σ·q·(n−m)

Coulomb's Force The general form of the Coulomb's Law, when two electrically charged point-objects, A and B, are involved, is F = k·q_{A}·q_{B}/ R² where F is the magnitude of the force of attraction (in case of opposite charges) or repelling (in case of the same type of charge, positive or negative) in newtons(N) q_{A} is electric charge of point-object A in coulombs(C) q_{B} is electric charge of point-object B in coulombs(C) R is the distance between charged objects in meters(m) k is a coefficient of proportionality, the Coulomb's constant, equals to 9.0·10^{9} in N·m²/C²

We have introduced a concept of electric field intensity as a force acting on a probe point-object B, charged with +1C of electricity, from a field produced by the main object A. This force is a characteristic of a field at a point where a probe object is located and is equal to E = k·q_{A}/ R²

If we want to move a probe point-object from one point in the field to another in uniform (without acceleration) motion, we have to take into account this force. It can help us to do the move, if this force acts in the direction of a motion, or prevent this motion, if it acts against it. In a way, the electric field becomes our partner in motion, helping or preventing us to do the move.

Work of Coulomb's Force Of obvious interest is the amount of work needed to accomplish the move. If we act against the force of electric field intensity, we have to spend certain amount of energy to do the work. If the field force helps us, we do not spend any energy because the field does it for us. Similar considerations were presented in the Gravitation part of this course.

Recall from the Mechanics part of this course that the work of the force F, acting at an angle φ to a trajectory on the distance S, is W = F·S·cos(φ)

For a non-uniform motion and variable force all components of this formula are dependent on some parameter x, like time or distance: dW(x) = F(x)·dS(x)·cos(φ(x)) where we have to use infinitesimal increments of work dW(x) done by force F(x) on infinitesimal distance dS(x). The angle φ(x) is the angle between a vector of force F(x) and a tangential to a trajectory at point S(x).

Using a concept of scalar product of vectors and considering force and interval of trajectory as vectors, the same definition can be written as dW(x) = (F(x)·dS(x))

The latter represents the most rigorous definition of work. Integration by parameter x from x=x_{start} to x=x_{end} can be used to calculate the total work

W_{[xstart , xend ]} performed by a variable force F(x), acting on an object in a non-uniform motion, on certain distance S(x) along its trajectory, as the parameter x changes from x_{start} to x_{end}.

In case of a motion of an electrically charged object in an electrical field the force is the Coulomb's force. Let's analyze the work needed to move such an object in the field of electrically charged point-object from one position to another.

Case 1. Radial Motion Let the charge of the main point-object in the center of the electrical field be Q. We move a probe object of charge q along a radius from it to the center of a field from distance r_{1} to r_{2}. The Coulomb's force on a distance x from the center equals to F(x) = k·Q·q / x² The direction of this force is along the radial trajectory and the sign of the Coulomb's force properly describes whether the resulting work will be positive (in case of similarly charged main and probe objects, + and + or − and −) or negative (in case of opposite charges, + and − or − and +).

Since the force is variable and depends on the distance between the main object and the probe object, to calculate the work needed to move the probe object, we have to integrate the product of this force by an infinitesimal increment of the distance dx on a segment from r_{1} to r_{2}. W_{[r1,r2]} = ∫_{[r1,r2]}k·Q·q·dx / x² Since the indefinite integral (anti-derivative) of 1/x² is −1/x, the amount of work is W_{[r1,r2]} = k·Q·q·(1/r_{1}−1/r_{2}) This work is additive. If we move from a distance of r_{1} to a distance r_{2} and then from a distance r_{2} to a distance r_{3}, the total work will be equal to a sum of works, which, in turn, would be the same as if we move directly to distance r_{3} without stopping at r_{2} W_{[r1,r2]} + W_{[r2,r3]} = W_{[r1,r3]}

Case 2. Circular Motion Consider now that we move a probe object circularly, not changing the distance from the center of the electric field. In this case the vector of force (radial) is always perpendicular to the vector of trajectory (tangential). As a result, this motion can be performed without any work done by us or the field. So, for a circular motion the work performed is always zero. Obviously, this work, as we move the probe object circularly, is also additive.

Case 3. General Any vector of force in a central electric field can be represented as a sum of two vectors - radial, that changes the distance to a center of a field, and tangential (along a circle), which is perpendicular to a radius. That means that any infinitesimal increment of work can be represented as a sum of two increments - radial and tangential. Since the latter is always zero, the amount of work performed to facilitate this motion depends only on the distances to the center at the beginning and at the end of the motion.

Conservative Forces The immediate consequence from this consideration is that the work needed to move a charged point-object from one point in the radial electric field to another is independent of the trajectory and only depends on starting and ending position in the field. Even more, for radial electric field it depends only on starting and ending distances to a center of the field.

This independence of work from trajectory is a characteristic not only of radial electric field, but of the whole class of the fields - those produced by conservative forces, and electrostatic forces are conservative. Gravitational forces are also of the same type. As an example of non-conservative forces, consider an object moving inside the water from one point to another. Since the water always resists the movement, the longer the trajectory that connects two points - the more work is needed to travel along this trajectory.

Electric Field Potential The electric field potential is a quantitative characteristic of an electric field, defined for each position in this field, as the amount of work needed to move a probe point-object of +1C from infinity (where the field does not exist) to this position in the field.

In the radial field produced by the point-object charged with Q amount of electricity the electric field potential at any point in the field depends only on a distance of this point to a center of the field. Using the formula above for r_{1}=∞ and r_{2}=r for a probe object charged with q=+1C of electricity, we obtain the formula for an electric potential at distance r from a center of the field, where a point-object charged with Q amount of electricity is located V(r) = −k·Q / r

Notice that the derivative of potential V(r) by distance r from a center gives the field intensity: V'(r) = k·Q / r² So, knowing the potential at each point of the radial field, we can determine the intensity at each point.

Since, as we stated above, the work performed to move a probe object in the electric field does not depend on trajectory, we can accomplish moving a probe object charged with +1C of electricity from distance r_{1} to distance r_{2} by, first, moving it to infinity, which results in amount of work W_{1} = −V(r_{1}), then from infinity to distance r_{2}, which results in amount of work W_{2} = V(r_{2}). The sum of them for a probe object charged with +1C of electricity gives the same amount of work as to move it directly from distance r_{1} to r_{2} as calculated above: W = W_{1} + W_{2} = −V(r_{1})+V(r_{2}) = = k·Q·q·(1/r_{1}−1/r_{2}) = W_{[r1,r2]}

Electric potential for each point of an electric field fully defines this field. If we know the electric potential in each point of a field, we don't have to know what kind of an object is the source of the field, nor its charge, nor shape.

To find the amount of work needed to move a charge q from a point in the electric field with a potential V_{1} to a point with potential V_{2} we use the formula W = q·(V_{2}−V_{1})