Unizor - Physics4Teens - Electromagnetism - Magnetic Field - Lorentz Force

Notes to a video lecture on http://www.unizor.com



Magnetism - Lorentz Force



In this lecture we will look at the interaction between an electric current and a magnetic field.



We start with an analogy between magnetic properties of a wire loop with
electric current running through it and those of a permanent magnet.
This have been explained in the previous session from the position of Ampere model of magnetism.

The picture below illustrates this analogy.



The wire loop with electric current running through it (thin black arrow
from left to right) creates a magnetic field around it. The lines of
this magnetic field (thin dark blue arrows from bottom up) go through
the wire loop and around it, closing on themselves, forming their own
loops. Inside the wire loop the direction of magnetic lines is from
South pole to North, while outside the wire loop they go from North pole
to South. Those magnetic filed line loops that are on the same distance
from the wire make up a tubular surface (a torus) around the wire.



This wire loop with electric current running through it and a magnetic
field around it would behave like a magnet, like a compass arrow, for
example.



In particular, positioned inside some external magnetic field, like in
the magnetic field of the Earth, and allowed to turn free, it will
orient itself in such a way that its North pole will point to a South
pole of an external magnetic field, which, in case of the magnetic field
of the Earth, is located not far from its geographical North pole.



It should be noted that circular form of a wire loop is not essential.
If it's rectangular, the magnetic behavior will be the same. It is,
actually, more convenient to work with a rectangular frame to illustrate
the interaction of magnetic field and electric current.



Let's start the experiment with a rectangular wire loop, that can rotate
around a vertical axis in the external magnetic field. Position it such
that one vertical segment of a wire is close to one pole of an external
magnet, while an opposite side is close to another pole. Let the
electric current run through it.



If this wire frame with electric current running through it is allowed
to rotate around a vertical axis, it will reorient itself with its North
pole directed to the left towards the South pole of an external magnet,
and its South pole directed to the right towards the North pole of an
external magnet, as shown on a picture below. The distance from the
North pole of an external magnetic field to both vertical sides of a
wire will be the same. Same about the South pole of the external
magnetic field.



This turn of a wire is, obviously, the result of forces of interaction
between external magnetic field and electric current with its own
magnetic field around it.



Consider the same two states of a wire (before and after the turn) viewed from above.

The initial position of a wire, viewed from above with rotating forces acting on it (blue arrows) is



Magnetic field lines of a wire with electric current running through it
are oriented along vertical direction on this picture, while the
magnetic field lines of an external magnetic field are horizontal.



As in the case of a compass arrow, aligning itself along the magnetic
field lines of the Earth, the external magnetic field forces, acting on
the magnetic field of a wire (blue arrows), will turn the wire to orient
its magnetic field lines along the magnetic field lines of an external
magnetic field, as shown on the following picture





At this final position external magnetic field forces (blue arrows) are
balancing each other and the rotation of a wire (after a short wobbling)
will stop.



Let's analyze the forces acting on a wire to turn it this way.

For this we don't really need a wire loop of any shape, it's sufficient
to have a linear wire with electric current running through it
positioned in an external magnetic field.



If we open up a wire loop into a straight line with electric current
running through it, the magnetic field around a wire will still exist,
and its lines will be positioned around a wire. Magnetic lines located
on the same distance from a wire with electric current will form a
cylinder with the line of electric current being its axis.



The picture below illustrates the force acting on a straight wire with
electric current running through it (straight black line) and its own
magnetic field (thin orange ovals around a wire) when it's positioned in
the external magnetic field. In this case the lines of the external
magnetic field (light blue arrows going left to right) are perpendicular
to the wire and the direction of the force is perpendicular to both,
the direction of the current in the wire and the direction of the
magnetic lines of the external magnetic field.



The direction of the force can be determined by the "rule of the right
hand", which states that, if the magnetic lines of the external magnetic
field are perpendicularly entering the right hand, while the thumb is
directed towards the electric current in the wire, fingers will show the
direction of force.

A different formulation of the "right hand rule" that results in the
same configuration states that, if magnetic field lines of the external
magnetic field are positioned along the fingers in the direction pointed
by them and the electric current in the wire is running in the
direction of the thumb, then the force exerted by the magnetic field on
the wire is perpendicular to the hand going outside of it.



When the wire has a rectangular shape, as on the picture in the
beginning of this lecture, two side of a rectangle are perpendicular to
the magnetic lines of an external magnetic field. The current in these
wire segments is running in opposite directions. As a result, the force
of magnetic field pushes these sides of a wire in opposite directions,
and the wire will turn until these two opposite forces balance each
other.



The force of an external magnetic field exerted on the wire with electric current running through it is called Lorentz force.



All the above considerations on interaction between an external magnetic field and an electric current are of qualitative character.

Let's address quantitative character of this interaction.



For starter, we will reduce our interest only to a case of a uniform
magnetic field and an infinitesimally thin straight wire running
perpendicularly to the magnetic field lines, like on the picture above.



It is reasonable to assume that the force exerted by a magnetic field acts on each moving electron within a wire.



Considering the force does not exist, if there is no electric current in
a wire (electrons are not moving), but can be observed only when there
is an electric current in a wire, another reasonable assumption is that
the force depends on the speed of moving electrons, which can be
measured as amperage of the electric current.



Experiment shows that the force is proportional to an amperage,
which can be intuitively explained by the idea that the higher the
amperage - the greater "number" of magnetic field lines of an external
magnetic field, crossed by electrons per unit of time, and each such
crossing results in certain incremental increase in the force exerted by
a field.



One more natural assumption is that the longer the wire - the
proportionally greater is the force exerted on it by a magnetic field.
This also is related to the above mentioned idea of a magnetic field
exerting a force on each electron crossing its magnetic field lines.



As a result, we come to a conclusion that the force is proportional to a product of electric current and the length of a wire:

F = b·I·L

where

b is a coefficient of proportionality that characterizes the strength of an external magnetic field,

I is the amperage of an electric current running through a wire,

L is the length of a wire.



What's interesting about this formula is that it allows to establish the
units of measurement of the strength of a magnetic field in terms of
units of measurement of force (F), electric current (I) and length (L).



DEFINITION

A uniform magnetic field that exerts a strength of (1N) on a wire of

1 newton1 meter (1m) length with a current running through it perpendicularly to the magnetic lines of a field of 1 ampere (1A) has a strength of 1 tesla (1T).

Tesla is a unit of measurement of the strength of a magnetic field.

The strength of a magnetic field is denoted by a symbol B. The Lorentz force is, therefore, expressed as

F = I·L·B

where

I is the amperage of an electric current running through a wire,

L is the length of a wire,

B is the strength of an external magnetic field



All the above considerations are valid for a case of an electric current
running perpendicularly to lines of a uniform magnetic field.



As mentioned above, the Lorentz force exerted on a wire depends on the movement of electrons in the wire crossing the magnetic lines of an external magnetic field.

Simple geometry prompts us to conclude that, if the direction of the
current is not perpendicular to magnetic lines of an external magnetic
field, but at angle φ with them, the number of magnetic
lines crossed by electrons in a unit of time is smaller and, actually,
is smaller by a factor sin(φ).



So, for any angle φ between the electric current and magnetic field lines of an external magnetic field the formula for Lorentz force would be

F = I·L·B·sin(φ)

where

I is the amperage of an electric current running through a wire,

L is the length of a wire,

B is the strength of an external magnetic field

φ is the angle between the direction of the electric current and lines of an external magnetic field



Taking into consideration the direction of the Lorentz force
perpendicular to both vectors - electric current (from plus to minus)
and lines of an external uniform magnetic field (from South to North),
the above formula can be represented using a vector product

F = I ·L⨯ B  

Unizor - Physics4Teens - Electromagnetism - Magnetic Field - Inside Magnet

 



Notes to a video lecture on http://www.unizor.com



Magnetism - Internal Structure

of Magnets



Let's look inside a permanent bar magnet with two poles, North and South.

We model its magnetic properties as a result of a cumulative properties
of individual electrons rotating along parallel axes within parallel
planes in the same direction.

Each such rotating electron represent a tiny magnetic dipole with
its own North and South poles with attracting force between opposite
poles (North and South) and repelling force between the same poles
(North to North or South to South).



The attraction between two rotating electrons that face each other by
opposite poles we have explained by the fact that in this case electrons
rotate in the same direction and "help" each other. The repelling of
two rotating electrons that face each other by the same poles is
explained by the fact that they rotate in opposite directions and
"disturb" each other.



Since we are talking about permanent magnet, all axes of rotation of
electrons are always parallel to each other and planes of rotation are
always parallel as well.



Consider a situation of two electrons rotating on parallel planes around the same axis on the same radius.



In this case the magnetic properties of the South pole of the upper (on
this picture) electron are neutralized by properties of the North pole
of an electron under it.

So, the magnetic field of a pair of electrons in this position is the
same as for one electron with poles located on a greater distance from
each other.



Now expand this logic to a full size of a bar magnet. The result is that
all internal connections between South and North poles will neutralize
each other and the only significant magnetic properties are of those
electrons concentrated on two opposite surfaces of a magnet where its
North and South poles are located.



This looks like some magnetic charges of opposite types, that we called North and South, are concentrated on two opposite ends of a magnet.

These magnetic charges behave similarly to electric charges,
except magnetic ones always come in pairs. We can even think about
magnetic equivalent of the Coulomb Law. The only complication is that we
always have a superposition of two magnetic fields coming from two ends
of magnetic dipole.

This is the Gilbert model of magnetic properties, attributed to
William Gilbert, an English physician (including a physician for English
royalty), who published in 1600 a six volume treatise that contained
all the information about electricity and magnetism known at that time.
Gilbert was the one who discovered magnetic properties of Earth and came
up with formulation of properties of magnets and terminology that
describes them (like magnetic poles).



Consider a different approach - two electrons rotated within the same
plane around parallel axes and immediately near each other. The common
plane of rotation is, of course, perpendicular to the magnet's
North-South axis and axes of rotation of these electrons are parallel to
the magnet's North-South axis.



Electrons moving near each other are moving in opposite directions and
neutralize each other, as if there is no current there at all. So,
within every plane perpendicular to the North-South axis of a magnet all
inner currents are neutralized, and the only really present current is
around the outer boundary of a magnet.



This is the Ampere model of magnetism. It makes the magnetic
properties of permanent magnet equivalent to properties of an electric
current in a loop around the side surface of a magnet with each electron
moving within a plane perpendicular to a magnet's North-South axis.


This model of magnetism is extremely important, as it connects the
magnetic properties to those of properties of electric current and shows
inherent connection between electricity and magnetism.



It also opens the door to electromagnetism - generating magnetic field using electricity.

A loop of electric current acts similar to each electron inside a
permanent magnet, just on a larger scale. A number of electric current
loops of the same radius around the same axis parallel to each other
makes the magnetic field even stronger.





If we make a loop of electric current and put an iron cylinder (which by
itself does not have magnetic properties) inside this loop, the iron
cylinder will become magnetic, and the more loops the electric current
makes around this cylinder - the stronger the magnetic properties of an
iron cylinder will be, and it will act exactly as the permanent magnet,
becoming electromagnet.

But, as soon as we stop the flow of electric current around this cylinder, it will lose its magnetic properties.



Another important feature of the Ampere model is that it allows
to measure the strength of the magnetic field produced by an
electromagnet by such known physical quantities as amperage of the current circulating in the wire loops, producing the magnetic field, and some geometric properties of the wire loops.

Unizor - Physics4Teens - Electromagnetism - Magnetic Field

Notes to a video lecture on http://www.unizor.com



Magnetism - Magnetic Field



Magnetic forces act on a distance, so there must be a field that exists around each magnet - magnetic field.



When studying Electrostatics, we started with the simplest
electrically charged object - a point-object with certain excess or
deficiency of electrons to make it negatively or positively electrically
charged.

The electrostatic field around it was spherical in shape and the only
important parameter that determined the relative position of a probe
object (also a point-object, positively charged with one coulomb of
electricity) was a distance of this probe object from the source of
electrostatic field.



If we wanted to analyze the electric properties of a more complex source
of electricity, like a rod or a sphere, we could always resort to some
relatively simple geometry and calculus to achieve our goal by breaking a
larger object into smaller parts.



Looking at the electrostatic field, the forces acting on a probe object
are always unidirectional, either attracting or repelling. To calculate
the resultant force, we used a vector sum of them, and in simple cases
of sources of electrostatic fields (a point-object, a rod, a sphere) it
was relatively simple task.



With magnets the situation is much more complex. We cannot have a
point-object because each magnet has two poles and each of them act in
some way. Even more, the magnetic force exhorted by a magnet is changing
as we move from North pole to South, first diminishing to zero in the
middle and rising again at the other pole.



Probably, the simplest magnet we can deal with, as a source of magnetic
field and a probe object, is a thin rod with poles at its ends. But even
in this case we have to take into account all the different forces,
attracting and repelling, of different magnitude and directions that act
on probe magnet.



Let's experimentally visualize the magnetic field in this simplest case.

For this experiment we need a bar magnet and iron shavings. Each shaving
is a little temporary magnet that forms its poles based on the forces
of the magnetic field. Then different shavings will attach to each other
by opposite poles and form lines. The picture obtained will represent
the vectors of forces in the magnetic field of a bar magnet. Along each
line on a picture below lie iron shavings, each with North and South
magnetic poles, linked by opposite poles and directed with their South
pole closer to North pole of a bar magnet.



The picture below schematically represents these field forces.



Direction of forces from North magnetic pole to South was traditionally
chosen, similarly to a direction of the electric current was chosen from
a positive terminal to a negative one, regardless of the flow of
electrons, unknown at the time of early experiments with electricity.



Let's discuss a concept of a magnetic field.

As usually, the following explanation is the model, which to some
degree corresponds to experimental and theoretical data, but we do not
claim that in reality the things are arranged in exactly this way.

However, we offer it as an aid to understanding the concept of a magnetic field.



Imagine a small particle rotating within a plane with certain speed on a
certain radius around an axis that is perpendicular to this plane. On
the same axis in a plane parallel to the first one another particle is
rotating. Generally speaking, it might rotate on a different radius, in
the same or opposite direction and with a different speed.



Consider a distance between these two parallel planes of rotation of
these two particles. When it's large, particles don't really have any
interaction. But, when we make this distance small enough, the particles
will "feel" each other.



If the particles rotate in the same direction, there will be some
attracting force between them and the planes of rotation tend to get
closer to each other.

If the particles rotate in the opposite directions, there will be some
repelling force between them and the planes of rotation tend to increase
the distance between them.



The rotating particles in this model behaves like a bar magnets
positioned along the axis of rotation with poles determined by a
direction of rotation. We can assume that the North pole is defined by a
rule, that, looking from it towards the rotating particle, this
particle rotates counterclockwise.

If two particles rotate around the same axis in the same direction, each
behaving like a bar magnet, the magnets will attract to each other
because they will be facing by opposite poles.

If the particles rotate in opposite direction, the corresponding magnets
will be facing each other by the same pole and will repel each other.



Now let's assume that these particles are electrons rotating around a
nucleus in the atoms inside some object. According to this model, if all
planes of rotation of electrons inside all atoms are parallel and the
rotation is such that all North poles of all atoms are directed in the
same way, we have a perfect permanent magnet.



If planes of electron rotation in some object are randomly directed and
are insensitive to outside forces exhorted by magnets, we have a
diamagnetic object - the one that cannot be magnetized.



If planes of electron rotation in some object are randomly directed, but
outside magnetic forces, interacting with atoms of this object, align
all planes of rotation in a parallel fashion with the same direction of
the poles, we have an object that can be a temporary magnet.



In all those cases our model of rotating electrons, producing a force
that in some way acts like a bar magnet positioned along the axis of
rotation, seems to explain all the magnetic properties.

It also prompts to a connection between the electricity and magnetism
because both are caused by position and movement of electrons.

Unizor - Physics4Teens - Magnets - Practical Aspects

Notes to a video lecture on http://www.unizor.com



Magnetism - Practical Aspects



Here is a list of a few cases where permanent magnets of different kind are used.



A magnetic screwdriver.

Refrigerator door.

Magnetic letters.

Bracelets with magnetic closure.

Compass.

Magnetic toys.

Different machinery uses magnets (e.g. large disk magnets to pick up metal junk).

Medical instruments (e.g. MRI - Magnetic Resonance Imaging).

In phones and headphones.

In all kinds of motors and generators.

In televisions.

In computers.



Permanent magnets occur in nature. A mineral magnetite is such a permanent magnet. It's black and shiny, when polished. The chemical composition of magnetite is iron oxide, which means that its molecule contains only atoms of iron and oxygen, the formula is Fe₃O₄.

Two pieces of magnetite will attract each other if faced by
opposite poles (North of one to South of another) and repel each other
when positioned facing each other by the same pole.



Magnetite attracts certain metals, like iron.



A piece of magnetite, if hanging on a thread or free floating in
water on a wooden plate will always turn along a meridian, thus people
used it in navigation before they invented a compass.



The naturally occurring magnetite is quite a weak magnet. Much
stronger permanent magnets are created artificially by melting together
certain metals. One of the strongest artificial magnets are made by
combining iron, neodymium and boron into an alloy called neodymium magnets.



Artificial magnets can have any shape with location of the poles chosen
by a manufacturer. For example, a ring magnet can be created with a
North pole on the outside circumference of a ring and South pole on the
inside circumference.



Consider a bar magnet with poles on its opposite sides.

Let's use an iron nail as a probe object to measure the force of attraction at different points on a magnet.

The strongest force of attraction will be at the ends of a bar magnet,
at its poles. If we move a nail from one end of the magnet to another,
the strength of attraction will diminish to zero at the middle, then
again will rise to its maximum on an opposite pole.

The midpoint of a bar magnet has no attracting strength at all.



Interestingly, if we break a bar magnet in the middle, each half will
not be a single pole magnet, single pole magnets don't exist. Each half
will have two poles, one old and the new one at the point we broke the
original magnet. Each new smaller magnet will be weaker than original.





Artificial magnets can be permanent or temporary.
Permanent magnets do not change their magnetic properties. Temporary
ones can be magnetized, when are near another permanent magnet, or
demagnetized, when there is no other magnets nearby. Plain nails and
paperclips have such a property. Their polarity depends on the position
of the other permanent magnet. Another example is electromagnets that
will be discussed later on in the contents of a connection between
electricity and magnetism.



There are artificial magnets that, after being magnetized by electric
impulse, retain their magnetic properties, including the polarity. At
the same time, exposed to another impulse, they can lose their magnetic
properties or change the polarity. This type of magnets were used in the
memory of the first computers with each bit of information (1 or 0)
stored as a magnetization of one ferrite ring.





Artificial magnets can be not only in some solid shape, but also form a
thin magnetic layer on a surface of some other material. This is how
magnetic tape and magnetic disks were created, both used extensively in
the computers. The principle of work of these memory devices is based on
the properties of the thin magnetic layer to retain the state of tiny
magnets this layer contains. Special devices can "write" information on
the magnetic layer by temporary magnetizing the tiny magnets of the
layer and "read" this information.



Another example is injecting a magnetically-sensitive liquid into a
tumor and, using a powerful magnet, heat it up to destroy the tumor.



The fact that permanent magnets are permanent and, seemingly, represent
an unlimited source of energy, prompted many people to construct a
device that would exploit this property to create a perpetual movement
and use it to generate unlimited amount of power.



Some of these devices did move for a long time. However, they contradict
the principle of energy conservation for a closed system and, sooner or
later, friction and other forces would stop it.

Here is a good example of such a device



https://www.youtube.com/watch?v=UqlPePYgdg4

Unizor - Physics4Teens - Electromagnetism - Magnets

 



Notes to a video lecture on http://www.unizor.com



Magnetism - Magnets



We usually don't have a lot of questions about forces that act on a contact. There is an obvious cause and observable effect.



Forces that act on a distance cause a logical question "Why?" since
there is no obvious cause for observable actions, like attraction or
repelling of objects without immediate contact between them.



So far, we studied gravitational force, acting on a distance, and electrostatic force, also acting on a distance.



We have introduced a concept of a field to explain these actions on a distance and quantitatively study them.



In case of gravity we just stated that all objects have certain characteristic, mass, and any object has a gravitational field
around it that acts on any other object in this field, according to the
Law of Gravity. The force is always attracting and proportional to a mass. The gravitational force always exists around any object.



In case of electrostatics we explained the existence of a force on a distance and electrostatic field by an excess or deficiency of electrons, which we called an electric charge and assigned negative charge (−) to an excess of electrons and positive charge (+)
to their deficiency. In this case the force between two objects
positioned at a distance from each other can be either attracting (if
one object is positively charged and another - negatively, opposite
attracts) or repelling (for similarly charged objects, similar repels)
and is proportional to the number of electrons in excess or deficiency.



The electrostatic force exists only if there is an excess or deficiency
of electrons that usually requires certain effort to create by
separating certain electrons from their atoms. Electrostatic forces are
usually much stronger than gravitational.



There is another observable force on a distance. Certain objects, which are called permanent magnets,
have a force field around them that does not require any efforts to
create. This makes them similar to gravitation. But, while the force of
gravity is always attracting, magnets attract only some objects - those
that have certain magnetic properties.



The force field around magnets always exists and has an interesting property - polarity. Each magnet has two poles, which are conditionally called North and South.



Two magnets attract each other, if they are facing each other with
different poles, that is North pole of one magnet attracts South pole of
another (again, opposite attracts). Positioned to face each other by
the same pole (North to North or South to South), they repel each other.



Another interesting fact that differentiates magnets from electrically
charged objects is that magnets always have two poles - North and South.
There is no magnet with only North or only South pole. In the world of
electricity objects are usually either positively charged (deficiency of
electrons) or negatively charged (excess of electrons), not both at the
same time, but in the world of magnetism existence of two poles, North
and South, is a necessary property of every magnet.



Interestingly, our planet Earth is a giant magnet with North magnetic
pole positioned close to the geographical North pole (but not exactly at
it) and South magnetic pole positioned close to South geographical
pole. North and South magnetic poles of Earth are, actually, moving
because of movements of the planet core.



Compass is a device with a small magnet in a form a free rotating arrow,
also having North (usually, blue) and South (usually, red) poles. Since
opposite poles attract, the red part of a compass arrow (South pole of
an arrow magnet) points to the North magnetic pole of Earth.





In places far from Earth poles we can rely on a compass to determine the
general direction to the North. Close to the geographical poles a
compass is not a reliable tool, because the North and South magnetic
poles of Earth do not coincide with its geographical poles.



We do know that electric charge is caused by excess or deficiency of
electrons. This is a relatively superficial explanation. We never
discussed what caused two electrons to repel each other or proton and
electron to attract each other. These are much deeper issues that
require significantly more study.

With magnetism we also need some, however incomplete, explanation for this phenomenon.



Here is what might serve as a superficial explanation.



Electrons inside the atoms have their own magnetic properties, derived from their movement and so-called spin,
studied in Quantum Theory. Pair of electrons usually has electrons with
opposite spins and their magnetic poles neutralize each other. But in
case of odd number of electrons this balance is distorted. Atom with
such a magnetic unbalance becomes a source of magnetic field.



Another necessary property of the magnet is the relatively
unidirectional position of these atoms (all North poles are pointing to
the same direction), thus making the effect of magnetism much stronger.

Unizor - Physics4Teens - Electromagnetism - Direct Current - Electric He...

Notes to a video lecture on http://www.unizor.com



Direct Current - Electric Heat - Problems 1



Problem A

A lamp with resistance 500Ω (ohm) is powered by a source of electricity producing a voltage of 100V (volt).

What is the power consumption of this lamp?



Solution

Power is work per unit of time. It can be expressed as

P = U·I or

P = U² / R or

P = I²·R

Substituting the real numbers,

P = 100² / 500 = 20 (watt)



Problem B

Express the work performed by a source of electricity in Problem A during 8 hours in joules and in kilowatt-hours.



Solution

By definition,

1 watt = 1 joule/sec

Power consumed by a lamp is 20 watt (that is, 20 joules per second).

During 8 hours (that is, 8·3600 seconds) the work performed by a source of electricity will be equal to

W = 20·8·3600 = 576000 J (joules)

In kilowatt-hours (kWh) the work will be a product of power in kilowatts and time in hours, that is

W = 0.020·8 = 0.160 kWh



Problem C

There are usually two characteristics written on an incandescent
electric lamp: the difference in electric potential on the terminals of a
source of electricity the lamp is supposed to be connected to in volts (or voltage) and the power it consumes in watts (or wattage).

Let the voltage of a lamp be U=220V and the wattage - P=55W.

Determine the electric current I going through a lamp and its electrical resistance R.



Solution

Power is work per unit of time. It can be expressed as

P = U·I or

P = U² / R or

P = I²·R

From this, knowing U and P, we derive I and R.

I = P / U

R = U² / P

Substituting the real numbers,

I = 55 / 220 ≅ 0.25A (amperes)

R = 220²/55 = 880Ω (ohms)



Problem D

What electric current is supposed to go through an incandescent lamp with characteristics 110V, 55W, and what electric current will be going through it, if we connect it to a source of electricity with voltage 220V?

How much heat will be generated by this lamp in this case of abnormally high voltage?



Solution

P = U·I = I²·R = U²/R

R = U²/P

R = (110V)²/(55V) = 220Ω

I_norm = (55W)/(110V) = 0.5A

I_high = (220V)/(220Ω) = 1A

P_high = (220V)²/(220Ω) = 220W

In this case of high voltage the lamp will generate 220W of heat, which is 4 times greater amount of heat than normal. It will burn very soon.



Problem E

Two incandescent lamps are connected in series to a source of electricity with voltage 220V.

The first lamp has characteristic 220V and 55W.

The second lamp has characteristic 110V and 55W.

What electric current will be going through these lamps along a circuit?

Will the amount of heat produced by each lamp be greater or less than the one it's designed to produce?



Solution

P = U·I = I²·R = U²/R

R = U²/P

R₁ = 220²/55 = 880Ω

R₂ = 110²/55 = 220Ω

R_total = R₁ + R₂ = 1100Ω

I = U/R_total = 220/1100 = 0.2A

P₁ = 0.2²·880 = 35.2W

Consumed power of 35.2W is less than 55W, it will produce less light than it is manufactured for.

P₂ = 0.2²·220 = 8.8W

Consumed power of 8.8W is significantly less than 55W, it will be hardly lit at all.



Problem F

Two incandescent lamps are connected in parallel to a source of electricity with voltage 220V.

The first lamp has characteristic 220V and 55W.

The second lamp has characteristic 110V and 55W.

What electric current will be going through a common part of a circuit?

Will the amount of heat produced by each lamb be greater or less than the one it's designed to produce?



Solution

P = U·I = I²·R = U²/R

R = U²/P

R₁ = 220²/55 = 880Ω

R₂ = 110²/55 = 220Ω

R_total = 1 / (1/R₁ + 1/R₂) = 176Ω

I_common = U/R_total = 220/176 = 1.25A

P₁ = 220²/880 = 55W

Consumed power of 55W is exactly the same as normal, it will produce exactly the same amount of heat and light as it is manufactured for

P₂ = 220²/220 = 220W

Consumed power of 220W is 4 times greater than normal 55W, it will produced a lot of heat and light, but will burn soon.

Unizor - Physics4Teens - Electromagnetism - Electric Heat

Notes to a video lecture on http://www.unizor.com



Direct Current - Electric Heat



Consider a circuit with a source of electricity that produces a difference in electric potential (voltage) U(volts) between its terminals. The only other component in this simple circuit is a resistor with resistance R(ohms).



According to the Ohm's Law, the electric current going through a resistor is

I = U/R(amperes)



Recall that a difference in electric potential (voltage) of 1V(one volt) between two terminals of a source of electricity means that it requires one joule of work (1J) to move one coulomb of positive electric charge (+1C) between these terminals.

Therefore,

1V·1C = 1J and 1V = 1J/1C



Further recall that the electric current of 1A(one ampere) is the flow of electricity, when 1C(one coulomb) of electricity is moving across the wire within 1sec(one second).
Therefore,

1A = 1C/1sec and 1A·1sec = 1C



In case of our circuit with the voltage between the terminals of a source of electricity U, electric current I and resistor R we have I coulombs of electric charge going through a circuit each second.

So, it t seconds the source of electricity moves I·t coulombs of electricity.



Moving one coulomb (1C) of electricity between terminals with one volt (1V) of a difference in electric potential requires one joule (1J) of work.

Therefore, moving this one coulomb (1C) of electricity between terminals with a difference of electric potential (voltage) U volts requires U joules of work.

Continuing this logic, moving I·t coulombs of electricity between the terminals with the voltage U volts requires U·I·t joules of work.



So, the work performed by a source of electricity with voltage U (volts) that moves I coulombs electricity every second (amperage) through a resistor with resistance R ohms during time t sec is (in joules)

W = U·I·t



From the Ohm's Law U=I·R immediately follows

W = I²·R·t and W = U²·t / R



In most cases we will assume that wires have zero resistance and only
resistors have the property of resistance. So, in our simple circuit the
only resistance to the moving electrons is in a resistor R.

Since the source of electricity performs work, this work is supposed to
do something. According to the Law of Energy Conservation, the work is
just a transformation of one form of energy into another.

In our case the only result of the work performed by a source of
electricity is moving electrons in a circuit that, meeting resistance of
the atoms inside a resistor, push them around, thus increasing their
chaotic movement inside a resistor. This chaotic movement is heat that can be measured, for example, by measuring a temperature of a resistor.



The bottom line is that a source of electricity that has voltage V, producing an electric current with amperage I going through a resistor with resistance R during time t performs work

W = U·I·t = I²·R·t = U²·t / R

which is converted to heat that increases the temperature of a resistor.



This process is used in, for example, in incandescent lamps. The rising
temperature of a tungsten spiral inside such a lamp results in heat and
light radiation that lights and heats up the space around a lamp.