AC - Effective Voltage: UNIZOR.COM - Physics4Teens - Electromagnetism

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



Effective Voltage
and Current (RMS)



We all know the voltage in our outlets at home. In some countries it's
220V, in others 110V or some other. But we also know that electric
current distributed to consumers from power plants is alternating (AC).

The actual value of voltage and amperage are sinusoidal, changing with time as:

U(t) = U_max·sin(ωt)

I(t) = U(t)/R = I_max·sin(ωt)

where

ω is some parameter related to the generation of electricity at the power plant,

t is time,

U_max is the voltage amplitude,

I_max = U_max/R is the amperage amplitude,

R is resistance of the circuit where alternating current is running through.



If voltage is variable, what does it mean that in the outlet it is, for example, 220V?



The answer to this question is in comparing energy the electric current is carrying in case of a variable voltage with energy of a direct current.



The AC voltage is sinusoidal, so we can calculate the energy it carries through a circuit during the time of its period T=2π/ω and calculate the corresponding DC voltage that carries the same amount of energy during the same time through the same circuit.

That corresponding value of the DC voltage is, by definition, the effective voltage in the AC circuit, which is usually called just voltage for alternating current.



First, let's evaluate how much energy the AC with voltage U(t)=U_max·sin(ωt) carries through some circuit of resistance R during the time of its period T=2π/ω.



Consider an infinitesimal time interval from t to t+dt.
During this interval we can consider the voltage and amperage to be
constant and, therefore, use the expression of the energy flowing
through a circuit of resistance R for direct current:

dW(t) = U(t)·I(t)·dt = U²(t)·dt/R

(see "Electric Heat" topic of this course and "Ohm's Law" for direct current)



To calculate the energy going through a circuit during any period T=2π/ω, we have to integrate this expression for dW on an interval from 0 to T.

W_[0,T] = ∫_[0,T]dW(t) =

= ∫_[0,T]U²(t)·dt/R =

= ∫_[0,T]U²_max·sin²(ωt)·dt/R =

= (U²_max/R)·∫_[0,T]sin²(ωt)·dt



To simplify the integration we will use the known trigonometric identity

cos(2x) = cos²(x) − sin²(x) = 1 − 2sin²(x)

from which follows

sin²(x) = [1 − cos(2x)] /2



Next we will do the substitution

x = ω·t

t = x/ω

dt = dx/ω

t ∈ [0,T] ⇒ x ∈ [0,ωT]=[0,2π]



The integral above, expressed in terms of x, is

∫_[0,2π]sin²(x)·dx/ω =

= ∫_[0,2π][1−cos(2x)]·dx/(2ω) =

= [2π−∫_[0,2π]cos(2x)·dx]/(2ω) =

= π/ω

Therefore,

W_[0,T] = π·U²_max/(ω·R)



At the same time we have defined the effective voltage as the one that delivers the same amount of energy as W_[0,T] to the same circuit of resistance R during the same time T=2π/ω if the current is constant and direct.

That is

U²_eff·2π/(R·ω) = W_[0,T] = π·U²_max/(ω·R)



Cancelling and simplifying the above equality, we conclude

2·U²_eff = U²_max

or

√2·U_eff = U_max

For effective voltage 110V the peak voltage (amplitude) is 156V,

for effective voltage 120V the peak voltage (amplitude) is 170V,

for effective voltage 220V the peak voltage (amplitude) is 311V.

Properties of Alternating Current: UNIZOR.COM - Physics4Teens - Electrom...

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



Properties of

Alternating Current (AC)



Let's talk about practical aspects of electricity. We use it to light up
our homes, to move the motors that pump the water, to ignite the car
engine etc.

These are practical issues that must be resolved in the most efficient way.

A short statement about alternating electric current is that it allows to do things easier and more efficient than direct electric current.



The primary reason for using alternating electric current that allows to
perform all these functions more efficiently than using direct electric
current is that alternating electric current creates alternating magnetic field, that, in turn, can generate alternating electric current in another circuit through a mechanism of induction,
and at many points on the way of these transformations we can use some
device to extract some useful functionality, like rotate the motor.



Let's start from the beginning.

In order to use electricity, we have to generate it. In most cases we
convert mechanical movement into electricity at power plants. There are
other ways, like chemical reaction in the battery or solar panels, but
they produce limited amount of electricity currently used.



The most common way to transform mechanical motion into electricity is through induction. Recall, when we rotate a wire frame in the uniform magnetic field, the electric current is induced in the wire frame.



This process of generating electric current by transforming mechanical energy into electricity using the induction is at the heart of all power plants.

The simplest mechanical motion that can be used on industrial scale is
rotation. So, some mechanical source of energy can rotate the wire frame
in a permanent magnetic field, which generates the electric current,
that we can distribute through different circuits to different devices.



As we know, induced electromotive force (EMF) in a wire frame, rotating in a magnetic field, depends on the rate of change of magnetic flux through this frame. In a simple case of a uniform magnetic field of intensity B and a wire frame rotating with constant angular speed ω around an axis perpendicular to the direction of magnetic field lines the magnetic flux
going through a frame is changing with time, depending on the angle of
rotation of a wire frame and the direction of the vector of magnetic
field intensity.



The picture below represents the top view of a flat rectangular wire frame of dimensions a by b at some moment of time t, when the frame deviated by angle φ from its original position, which was parallel to the magnetic field lines.


As was calculated in a lecture about a rotation of a wire frame, the magnetic flux going through a frame is

Φ(t) = B·S·sin(ωt)

where S=a·b is the area of a wire frame.



The induced EMF U(t) in a wire frame equals to the rate of change of this flux:

U(t) = −dΦ(t)/dt = −B·S·ω·cos(ωt)

What's most important for us in this formula is that it shows the change of a magnitude and a direction of the induced EMF.

The dependence of U(t) on cos(ωt) shows that
the induced electric current in the wire loop and, consequently, in all
devices that receive this current from this wire loop and have combined
resistance R is changing from some maximum value I_max=B·S·ω/R down to 0, then it changes the direction to an opposite and grows in absolute value to I_max in that opposite direction, then again diminishes in absolute value to 0, changes the direction etc.



This electric current that increases and decreases in absolute values and changes the direction in sinusoidal way is called alternating electric current.



So, the most direct and effective way to produce an electric current causes this current to be alternating.



As a side note, which we might discuss later on in the course, it's
easier to rotate the magnetic field, keeping the wire frame steady than
described above for purely educational purposes. In this type of
arrangement the contacts between the wire frame and a circuit going to
consumers of electricity is permanent and, therefore, more reliable.



It is possible to produce direct electric current, transforming
rotation into induced electric current, but it involves more complex and
less efficient engineering. Even then the electric current will not be
perfectly constant, but will change between maximum and minimum values,
while flowing in the same direction.



The next step after generating the electricity is delivering it to
consumers. This requires an extensive distribution network with very
long wires and many branches.



The amount of energy produced by a power plant per unit of time is

P_gen = U·I

where

U is the output voltage

I is the output electric current

and all the above values are changing in time in case of alternating electric current.



The amount of energy that is delivered to a user of electricity is less
that this by the amount lost to overcome the resistance of wires that
deliver the electricity, which is per unit of time equals to

P_lost = I²·R

where

I is the electric current

R is the resistance of wires.



Using transformers, we can increase the voltage and, simultaneously, decrease the amperage
of an electric current. We will do it right after the electricity is
generated, which reduces the loss of energy in a wire that transmits
electricity on a long distance.



One of the important steps in the distribution of electricity is
lowering the voltage, because we cannot connect any normal electric
device, like a vacuum cleaner, to an outlet with hundreds of thousands
of volts.

This process of lowering the voltage is also easily accomplished using transformers.



As you see, transformers play an extremely important role in the
distribution of electric power. But the transformers work only when the
electric current is changing. So, the consideration of distribution of
electricity are also a very important reason why we use alternating
current.



Consider two solenoids, one, primary, connected to a source of an
alternating electric current, another, secondary, - to a circuit
without a source of electricity. Since the electric current in the
primary solenoid is alternating, the magnetic field inside and around it
will also be alternating.

We have derived a formula for generated magnetic field for infinitely
long solenoid, but, approximately, it is correct for finite ones too. In
this case the density of wire loops is proportional to a total number
of loops, since the length of a solenoid is fixed.

The formula for an intensity of a magnetic field generated by a solenoid is

B = μ₀·I·N

where

B is magnetic field intensity

I is electric current in a solenoid (alternating in our case)

N is density of wire loops in the solenoid (proportional to a number of wire loops for finite in length solenoid)

μ₀ is a constant representing the permeability of space.



According to this formula, magnetic field characterized by B will be sinusoidal because the electric current I is sinusoidal.

If our secondary solenoid is located inside the first one or close
nearby, the alternating magnetic field generated by the primary solenoid
will induce alternating EMF in the secondary one proportional to a number of loops in it, since each loop will be a source of EMF on its own.

If the number of loops in the secondary solenoid is smaller than in the
primary one, the induced EMF in the secondary solenoid will have smaller
voltage than in the primary one. This is the idea behind the transformers that are used to lower the voltage coming from the power plant.



Real transformers are built on the above principle with some
practical modifications that we will discuss in a separate lecture
dedicated to this topic.

What's important is that alternating electric current allows to change the voltage in the circuit for distribution purposes.



As we see, generation and distribution of alternating electric current is more efficient from practical standpoint than generation and distribution of direct electric current. That's why alternating electric current is so widely used.



Most important property of the alternating electric current is its sinusoidal dependency on time.

Voltage oscillates between −U_max and +U_max with certain frequency ω:

U(t) = U_max·cos(ωt) or

U(t) = U_max·sin(ωt)

(both formulas, based on sin(ωt) and cos(ωt) describe the same behavior and differ only in when is the start time t=0).



The maximum value U_max is called amplitude or peak voltage of an alternating current.



The time during which the value of U(t) changes from its positive peak to another positive peak (or from negative peak to another negative peak) is called a period of an alternating electric current.

If ω is the angular speed of rotation that caused the generation of alternating electric current, the period is, obviously, T=2π/ω.



Another characteristic of an alternating electric current is frequency, which is the number of periods per unit of time, which is, obviously, an inverse of a period, that is f=1/T=ω/2π and the unit of frequency is hertz (Hz), so the frequency of 50Hz means that the voltage changes from one positive peak to another 50 times per second..



The amperage behaves, according to the Ohm's Law, similarly, oscillating between −I_max and +I_max with the same as voltage frequency ω:

I(t) = I_max·cos(ωt) or

I(t) = I_max·sin(ωt)

(depending on when is the start time t=0)

where I_max = U_max / R and R is the resistance of the circuit.

Induced Variable EMF: UNIZOR.COM - Physics4Teens - Electromagnetism - La...

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



Induced Variable EMF



The most important fact about electromagnetic induction is that variable magnetic flux Φ(t) going through some wire frame generates in it an electromotive force (EMF) U(t) proportional to a rate of change of the magnetic flux and oppositely directed.

U(t) = −dΦ(t)/dt



Let's apply it to the following experiment.





The experiment consists of a permanent magnet moving into a wire loop,
as shown on the above picture, with different positions of a wire loop
relatively to a magnet at different times marked as t=1, t=2 etc.

At t=1 a magnet is still outside of the area of a wire loop, facing a loop with its North pole.

At t=2 a North pole of a magnet enters the area of a wire loop.

At t=3 a magnet covered half of its way through a wire loop, so the loop is positioned around a midpoint of a magnet.

At t=4 a South pole of a magnet leaves the area of a wire loop.

At t=5 a magnet is completely out of the area of a wire loop.



While the magnet is far from a wire loop (even before time t=1), practically no electricity can be observed in a wire loop. There are two reasons for it.

Firstly, the magnetic field is weak on a large distance.

Secondly, the direction of magnetic field lines in an area where a
wire loop is located at a large distance from a magnet is practically
parallel to the movement of a magnet (or movement of a wire loop
relative to a magnet). In this position a wire crosses hardly any
magnetic field lines and, consequently, the magnetic flux going through a wire loop is almost constant. Hence, the rate of the magnetic flux change is almost zero and no EMF is induced.



As we move the magnet closer (time t=1), its magnetic field in the area of a loop is increasing in magnitude and the angle between the magnetic field lines in the area of a wire loop and the direction of a wire loop relative to a magnet becomes more and more perpendicular. The magnetic flux going through a wire loop is increasing and, consequently, the EMF induced in a wire becomes more noticeable.



At time t=2 the magnitude of the magnetic field reaches
its maximum, the direction of magnetic field lines crossed by a wire
loop is close to perpendicular to a relative movement of a wire. The
magnetic flux changes rapidly and induced EMF in the wire is strong.



At time t=3 the wire loop is at the midpoint of a magnet.
The strength of a magnetic field at this position is very weak, magnetic
field lines are practically parallel to the relative movement of a
wire, and no noticeable EMF is generated.



At time t=4 the situation is analogous to that of t=2
with only difference in the direction of the magnetic field lines. Now
the wire is crossing them in the opposite direction, which causes the
induced EMF to have a different sign than at time t=2.



After that, at time t=5 the induced EMF is diminishing in magnitude and becomes almost not noticeable.



In the next experiment we replace the permanent magnet with a wire loop A
with direct current running through it. As we stated many times, the
properties of a wire loop with direct electric current going through it
are similar to those of a permanent magnet.



So, if we move the wire loop A with the direct current closer and closer to a wire loop B not connected to any source of electricity, the magnetic flux going through a wire loop B will be changing and, similarly to the first experiment, the induced EMF will be observed in this wire loop B.



Two positions of the loop B relative to a wire loop A are pictured above at times t=1, when wire loop A is very close to wire loop B, and t=2, when wire loop A has passed the location of wire loop B.



Similar considerations lead us to conclude that at times t=1 and t=2 the rate of change of magnetic flux through wire loop B will be the greatest and, consequently, the induced EMF in it will be the greatest.



However, these two EMF's will be opposite in direction because the
direction of crossing the magnetic lines will be opposite - at time t=1 magnetic lines are directed from inside wire loop B, but at t=2 the lines are directed into inside loop B.



In both experiments above the most important is that variable magnetic
flux going through a wire loop, not connected to any source of
electricity, induces the EMF in it.



Our third and final experiment is set as the second, except, instead of using constant direct current in the wire loop A and moving it through wire loop B to cause changing magnetic flux, we will keep the wire loop A
stationary, but will change the intensity of its magnetic field by
changing the magnitude of the electric current going through it.



The result of changing electric current I(t) going through wire loop A of radius R (as a function of time t) is changing magnetic field vector B(t) it produces.

We did calculate the value of the intensity of this field in the center of a loop as

B(t) = μ₀·I(t)/(2R)



If electric current I(t) is variable, magnetic field intensity vector B(t) is variable at all points around wire loop A, including points around wire loop B, if it's relatively close.

Therefore, the magnetic flux Φ(t) through wire loop B will be variable.

Consequently, the induced EMF

U(t) = −dΦ(t)/dt

will not be zero.



At this point it's appropriate to mention that, if wire loop A is a tight wire spiral of N loops, the variable magnetic field and, therefore, variable magnetic flux through wire loop B will be N times stronger.

That results in N times stronger EMF induced in wire loop B.



Analogously, if wire loop B is a tight wire spiral of M loops, the same EMF will be induced in each loop and, therefore, combined EMF will be M times stronger than if B consists of only a single loop.



The above considerations bring us to an idea of a transformer - a device allowing to change voltage in a circuit. This will be discussed in a separate lecture.

Solenoid: UNIZOR.COM - Physics4Teens - Electromagnetism - Magnetism of E...

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



Solenoid



Solenoid is a wire spiral like on this picture



It can be considered as a combination of individual circular wire loops.

We will consider an ideal case when the wire is infinitesimally thin,
the length of a solenoid along its axis is infinite and there is a
direct electric current running through it, that is running through each
of its loops.

Our purpose, as in a case of a single loop, is to determine the magnetic field inside such a solenoid.



Position our infinite in length solenoid along the X-axis such that the
X-axis is the axis of a solenoid. Let the radius of a solenoid's loops
be R and the density of the loops per unit of length along X-axis be N, that is, each, however small, segment dX on the X-axis contains N·dX loops.

Assume a direct electric current I is running through all loops.



Since a solenoid is infinite, magnetic field created by an electric
current running through it should be the same at any point on its axis.

So, let's calculate the intensity of a magnetic field B at the origin of coordinates. At any other point on the X-axis it will be the same.



The method of calculation the magnetic field is as follows.

Firstly, we will calculate the intensity of the magnetic field at
the origin of coordinates, generated by a single loop at the distance X from this point.

Secondary, we will take into account the density of the loops and
increase the calculated intensity of a single loop by a number of loops
on a segment from X to X+dX, assuming dX is infinitesimal increment.

The final step is to integrate the obtained intensity for all X from −∞ to +∞.



1. Magnetic field from a single loop at distance X from the origin of coordinates



Calculations of the magnetic field at any point on the axis going
through a wire loop is similar to but more complex than the calculation
of the magnetic field at the center of a loop we studied in the previous
lecture.

The picture below represents the general view of the wire loop with a point O,
where we want to evaluate its magnetic field and, below this general
view, the same loop as viewed from a plane this loop belongs to
perpendicularly to the X-axis.



As we know, the intensity OB of the magnetic field at point O, generated by an infinitesimal segment dS of a wire loop located at point on a loop A is perpendicular to both electric current I at point A and vector AO from a point on a loop A to a point of interest - the origin of coordinates O.



The direction of the electric current at point A is perpendicular to the plane of a picture. Therefore, vector of intensity OB is within a plane of a picture, as well as line AO. Now all lines we are interested in are on the plane of a picture and vector of magnetic field intensity OB at point of interest - the origin of coordinates O is perpendicular to a line from point A on a loop to a point of interest O:

AO⊥OB.



Let B' be a projection of the endpoint of vector OB onto X-axis. From perpendicularity of AO to OB follows that ∠AOP=∠OBB' and triangles ΔAOP and ΔOBB' are similar.



As we know, the magnitude of magnetic field intensity at point O
depends on the distance from the loop segment with electric current,
the length of this segment, the magnitude of the electric current in it
and a sine of the angle between the direction of an electric current in a
segment and the line from a segment to a point of interest:



Since the direction of the electric current is perpendicular to a plane
of a picture, the angle between the direction of an electric current in a
segment and the line from a segment to a point of interest is 90°, so
its sine is equal to 1, and the formula for intensity of a magnetic
field at point O produced by a segment dS at point A on the loop is

dB = μ₀·I·dS / [4π(R²+X²]



We should take into consideration only the X-component of the vector of
magnetic field intensity because the Y-component will be canceled by the
corresponding Y-component of a vector of intensity produced by a
segment of a loop at diametrically opposite to point A location.

Let k = sin(∠OBB').

Hence,

k = sin(∠AOP) = R/√X²+R².

Then the X-component of vector OB is

dB_x = μ₀·I·dS·k / [4π(R²+X²)] =

= μ₀·I·dS·R / [4π(R²+X²)^3/2]



To get a magnitude of the magnetic field intensity at point O from an entire loop we have to integrate the above expression by S from 0 to 2πR, and the result will be

B_x= μ₀·I·2πR² / [4π(R²+X²)^3/2]

Simplifying this gives

B(X) = μ₀·I·R² / [2(R²+X²)^3/2]

Incidentally, if our loop is located at X=0, the formula above gives

B_x = μ₀·I / (2R),

that corresponds to the results presented in the previous lecture for intensity of a magnetic field at the center of a loop.



2. Magnetic field from all loops located at distance from X to X+dX from the origin of coordinates



This is a simple multiplication of the above expression for B_x by the number of loops located at distance from X to X+dX from the origin of coordinates, which is N·dX, where N is a density of the loops in a solenoid.

The result is the intensity of a magnetic field at point O produced by an infinitesimal segment of our infinitely long solenoid from X to X+dX
μ₀·I·R²·N·dX / [2(R²+X²)^3/2]



3. Finding the magnetic field at any point inside a solenoid by integrating the intensity produced by its segment from X to X+dX by X for an entire length of a solenoid from −∞ to +∞



The above expression should be integrated by X from −∞ to +∞ to get a total value of the intensity of the magnetic field produced by an entire infinitely long solenoid at point O (or any other point on its axis).

B = γ·∫_{[−∞;+∞]} dX / (R²+X²)^3/2

where

γ = μ₀·I·N·R²/2



Substitute X=x·R in the integral with dX=R·dx.

Infinite limits for X will be infinite limits for x.

Then integral will be

∫_{[−∞;+∞]} R·dx / (R²+R²·x²)^3/2 =

= (1/R²)·∫_{[−∞;+∞]} dx / (1+x²)^3/2

The value of the integral can be easily calculated.

The indefinite integral is

x/√1+x² + C

So, the value of the definite integral between infinite limits is equal to

1 − (−1) = 2.



Therefore,

B = γ·2/R² = μ₀·I·N



IMPORTANT: The intensity of a magnetic field at any point on an axis of
an infinitely long solenoid does not depend on its radius.

Textbooks usually state that the intensity calculated above is the same
at all points inside an infinitely long solenoid, not only at the points
along its axis. The justification and the proof of this are beyond the
scope of this course.

Magnetic Flux: UNIZOR.COM - Physics4Teens - Electromagnetism - Magnetism...

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



Magnetic Flux



Consider a flat lying completely within an XY-coordinate plane closed
wire loop that includes a battery with direct electric current I running through it.



Since electric current generates a magnetic field around each however small segment of a wire dS
with circular magnetic field lines around it in the plane perpendicular
to this segment of a wire, the general direction of all magnetic lines
inside a wire loop will be parallel to Z-axis.

More precisely, vectors of magnetic field intensity B at
all points of XY-plane inside a wire loop are similarly directed
perpendicularly to this plane and parallel to Z-axis. They are not
necessarily equal in magnitude, though.



Consider an infinitesimal two-dimensional area of XY-plane inside a wire loop dA around some point P.
Since it's infinitesimal, we can assume that the intensity of the
magnetic field inside it is uniform and equal to the intensity at point P, which we will denote as vector B(P).

This intensity is a result of combined magnetic field generated at that point by all infinitesimal segments dS of a wire.

Each one of these components of a resulting intensity vector has a
direction perpendicular to a direction of electric current in segment dS (its source) and to a vector R from a chosen infinitesimal segment dS of a wire loop to point P, from which follows the perpendicularity of all vectors of magnetic field intensity to XY-plane and parallel to Z-axis.



Magnetic field, as other force fields, is additive, that is, the
combined intensity vector generated by an entire wire loop is a vector
sum of intensity vectors generated by each segment of a wire.



All these individual magnetic fields generated by different segments of a
wire are directed along the Z-axis, so we will concentrate only on
their magnitude to find the result of their addition.



Knowing the geometry of a wire and the electric current I in it, we can use the law we described in the previous lecture that determines the magnetic field intensity at any point P inside a wire loop generated by its any infinitesimal segment dS:

dB = μ₀·I·dS·sin(α) / (4πR²)

where

R is a magnitude of a vector R from a chosen infinitesimal segment dS of a wire loop to point P,

dS is the infinitesimal segment of a wire,

α is an angle between the direction of electric current in the segment dS and a vector R from this segment to a point P,

μ₀ is the permeability of free space.



Integrating this along the entire wire (this is integration along a curve) gives the value of magnetic field intensity B(P) at point P.



Now we assume that this process is done and function B(P) is determined, that is we know the magnitude B(P) of the intensity vector of magnetic field generated by our wire loop at any point P inside a loop.

Recall that the direction of this vector is always along Z-axis, that is perpendicular to a plane of a wire loop.



We define magnetic field flux Φ as a two-dimensional integral of a product of intensity B(P) by area dA.

In a simple case of a constant magnetic field intensity B(P)=B at any point P inside a loop this is a simple product of a constant magnetic field intensity B by the area of a wire loop A

Φ = B · A

If B(P) is variable inside a loop, the two-dimensional integration by area of the loop produces the magnetic field flux

Φ = ∫∫_AB(P)·dA



The complexity of exact calculation of magnetic field intensity B(P) at any point P inside a loop and subsequent calculation of the magnetic field flux
flowing through the wire loop are beyond the scope of this course, so
in our practice problems we will usually assume the uniformity of the
magnetic field with constant B(P)=B.



A concept of magnetic field flux is applicable not only for a
magnetic field generated by a wire loop with electric current running
through it. It is a general concept used to characterize the combined
effect of a magnetic field onto the area where we observe it.



Consider an analogy:

magnetic field intensity and amount of grass growing in the backyard in a season per unit of area,

wire loop area and area of a backyard,

magnetic field flux and how much grass growth in an entire backyard in a season.

Another analogy:

magnetic field intensity and productivity of each individual doing some work,

wire loop area and a team of people doing this work,

magnetic field flux and productivity of a team doing this work.



A general magnetic field flux is defined for a given magnetic field with known intensity vector B(P) at each point P of space and given two-dimensional finite surface S in space, through which magnetic field lines are going.



The purpose of our definition of magnetic field flux in this general case is to quantify the total amount of "magnetic field energy" flowing through a surface S.



Notice a small area around point P on a picture above.

If we consider it to be infinitesimally small, we can assume that
magnetic field intensity at any point of this small area is the same as
at point P.



If this small area is perpendicular to vector B(P), we would just multiply the magnitude of intensity vector B(P) in tesla by the area ΔA of a small area around point P in square meters and obtain the magnetic field flux ΔΦ going through this small area in units of magnetic flux -
weber = tesla · meter².

In a more general case, when this small area around point P is not perpendicular to magnetic field intensity vector B(P), to obtain the magnetic field flux going through it, we have to multiply it by cos(φ), where φ(P) is an angle between magnetic field intensity vector B(P) and normal (perpendicular vector) to a surface at point P.

So, the expression for an infinitesimal magnetic field flux around point P is

dΦ = B(P)·cos(φ(P))·dA



The unit of measurement of magnetic field flux is Weber (Wb) that is equal to a flux of a magnetic field of intensity through an area of

1 tesla (1T)1 square meter (1m²):

1 Wb = 1 T · 1 m²



Once we have determined the magnetic field flux flowing through a
small area around each point of a surface, we just have to summarize
all these individual amounts to obtain the total magnetic field flux flowing through an entire surface.

Mathematically, it means we have to integrate the above expression for dΦ along a surface S.



We will not consider this most general case in this course. Our magnetic fields will be, mostly, uniform and surfaces flat.

Problems on Self-Induction: UNIZOR.COM - Physics4Teens - Electromagnetis...

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



Problems on Self-Induction



Problem A



Consider a wire of resistance R₀ bent into a circle of a radius r with its ends connected to a battery producing a voltage U₀.

There is a switch that can turn the flow of electricity in this circuit ON or OFF during a short time period T.

When it turns ON, the resistance of the circuit, as a function of time R(t), changes from infinitely large value to the value of R₀ as

R(t) = R₀·T/t

When it turns OFF, the resistance changes from the value of R₀ to an infinitely large value as

R(t) = R₀·T/(T−t)

What is the EMF of self-induction U_i during the current switching ON or OFF?



Solution



Switching ON

According to the Ohm's Law, the electric current in the circuit is

I(t) = U₀ /R(t) = U₀·t /(R₀·T)

The magnetic field intensity inside a loop, as we discussed in the lecture on magnetism of a current in a loop, is

B(t) = μ₀·I(t) /(2r) = μ₀·U₀·t /(2r·R₀·T)

The magnetic flux going through the wire loop of the radius r is a product of the intensity of the magnetic field by the area of a loop

Φ(t) = B(t)·πr² = μ₀·U₀·t·πr² /(2r·R₀·T) = πμ₀·U₀·t·r /(2R₀·T)

The EMF of self-induction is the first derivative of the magnetic flux
by time with a minus sign (minus sign because the induced EMF works to
prevent the increase in the current)

U_i = −dΦ/dt = −πμ₀·U₀·r /(2R₀·T)

It's negative, so it reduces the overall voltage in the circuit and,
therefore, reduces the electric current in it. At the end of switching
ON (at t=T) the current will not reach its maximum value but will continue to rise after the switch is completely ON until it reaches it.



Switching OFF

According to the Ohm's Law, the electric current in the circuit is

I(t) = U₀ /R(t) = U₀·(T−t) /(R₀·T)

The magnetic field intensity inside a loop, as we discussed in the lecture on magnetism of a current in a loop, is

B(t) = μ₀·I(t) /(2r) = μ₀·U₀·(T−t) /(2r·R₀·T)

The magnetic flux going through the wire loop of the radius r is a product of the intensity of the magnetic field by the area of a loop

Φ(t) = B(t)·πr² = μ₀·U₀·(T−t)·πr² /(2r·R₀·T) = πμ₀·U₀·(T−t)·r /(2R₀·T)

The EMF of self-induction is the first derivative of the magnetic flux
by time with a minus sign (minus sign because the induced EMF works to
prevent the increase in the current)

U_i = −dΦ/dt = πμ₀·U₀·r /(2R₀·T)

It's positive, so it adds to the overall voltage in the circuit and,
therefore, increases the electric current in it, which, potentially, can
be harmful for electronic devices on the circuit.





Problem B



In the same setting as in Problem A calculate minimum time T_min of switching a circuit OFF in order for the current in a circuit grow no more then 10% from the theoretical value

I₀ = U₀ /R₀,I₀ = U₀ /R₀.



Solution



Induced EMF equals to (from Problem A)

U_i = πμ₀·U₀·r /(2R₀·T)

It's a constant and does not depend on time t.

The resistance of a circuit is growing with time from R₀ at t=0 to infinity at t=T. Therefore, the maximum electric current will be observed at t=0, when the resistance is minimal, and will equal to

I_max = (U₀+U_i) /R₀ = I₀ + (U_i /R₀)

In order for this current not to exceed I₀ by 10% the following inequality must be satisfied

U_i /R₀ ≤ 0.1·U₀ /R₀

The resistance cancels and the remaining inequality is

U_i ≤ 0.1·U₀

Using the value of U_i calculated in Problem A, we obtain the following inequality that should be resolved for time interval T

πμ₀·U₀·r /(2R₀·T) ≤ 0.1·U₀

Voltage U₀ cancels out and the remaining inequality for time interval T is

T ≥ πμ₀·r /(0.2R₀)



Practical case

Assume the following conditions

μ₀ = 4π·10⁻⁷ N/A²

r = 0.1 m

R₀ = 200 Ω

Then the time interval of switching OFF must be at least

T = πμ₀·r /(0.2R₀) ≅ 10⁻⁸ sec

That is a very short interval of time. Mechanical switches usually are
much slower, which means the electric current under normal usage in
homes will not significantly rise when we switch the electricity OFF.

IMPORTANT:

Check formula units

(N/A²)·m/Ω = N·m/(A²·Ω) =

[N·m → W → V·A·sec]

= (V·A·sec)/(A²·Ω) =

[Ω → V/A]

= (V·A·sec·A)/(A²·V) = sec

as it is supposed to be.

Self-Induction: UNIZOR.COM - Physics4Teens - Electromagnetism -

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



Self-Induction



Let's study the process of increasing the direct electric current in a
wire loop that is connected to a source of a direct electricity (like
battery) that generates electromotive force of voltage U₀.

This increase might be attributed to a decrease in resistance in this
loop. For example, if the wire loop was initially disconnected from a
source of electricity and we connected it, flipping some switch, the
resistance is changing from being practically infinite to some value R during a fraction of a second and the electric current would grow from zero to some value

I = U/R (Ohm's Law).





The first thing that we can observe is that there is a magnetic field
generated by an electric current with magnetic field lines going through
a loop, thus effectively making it a magnet.



The next thing we note is that the intensity of a magnetic field
generated by an increasing direct current is also increasing with the
current because of their relationship that we have described in the
lecture about magnetism of an electric current loop.

In the center of an ideally circular loop this intensity, as we calculated in that lecture, is

B = μ₀·I/(2R)

where

μ₀ is permeability of space,

I is the current in a wire loop,

R is the radius of a wire loop.



At other points around the wire the intensity of a magnetic field also
increases proportionally to the electric current in a wire. Even if the
shape of our wire loop is not ideally circular, the proportionality of
intensity of a magnetic field at any point to an electric current would
still be held, because each infinitesimal segment of a wire creates its
own magnetic field and the magnetic field intensity is an additive
function, that is, produced by two sources, intensities from each are
added as vectors, resulting in a combined intensity.

As a result, the magnetic flux going through a wire loop increases proportionally to an increase in electric current.



Now recall the Faraday's Law of magnetic induction. It states that changing magnetic flux going through an electric circuit generates electromotive force (EMF) proportional to a rate of change of the magnetic flux, where rate of change means a change per unit of time, which, using mathematical language, is the first derivative of a variable magnetic flux by time.



Let's ignore for now the signs of the rate of change of the magnetic flux and the EMF, then the Faraday's Law can be represented as

U = dΦ/dt

where

U is the absolute value of a magnitude of the generated EMF,

Φ is magnetic flux,

t is time,

dΦ/dt is the rate of change of magnetic flux (first derivative of magnetic flux by time), taken as an absolute positive value.



From

(1) the proportionality of a magnetic field intensity at any point around a wire to an electric current going through a wire,

(2) definition of magnetic flux and

(3) the Faraday's Law

follows that the increasing electric current, causing an increasing
magnetic flux going through a circuit (so, the first derivative of a
magnetic flux is positive), generates a secondary EMF in
the circuit that is proportional to a rate of change of a magnetic flux,
which, in turn, is proportional to a rate of change of the electric
current in a loop.

So, generated (induced) secondary EMF is proportional to a rate of change of an electric current in a loop.



This induced secondary EMF causes the secondary electric current in a loop that interferes with the original electric current.

The process of interference of two electric currents, varying original
one and that caused by a change of magnetic field flux, is called self-induction.



Now let's talk about the direction of that secondary current from the position of the Law of Energy Conservation.

If the secondary current flows in the same direction as the increasing
original one, the rate of change of electric current in a loop will
increase, causing a greater rate of change of the magnetic flux, causing
even greater rate of change of the electric current flowing through a
wire, causing even greater flux etc. This is a perfect source of free
energy, so it cannot be true.

Indeed, the secondary electric current must be directed opposite to an
increasing original one, thus slowing the rate of increase.

The EMF generated by self-induction is opposite to the one that originated the increase in the electric current in a loop.



As a practical implication of this principle, when we switch a lamp on,
and the time a switch actually closes the circuit is, for example 0.01
sec, the electric current in a circuit will not reach its maximum during
this exact time, but later, like during 0.02 sec, because of
self-induction that slows the increase of amperage.



Let's consider the process of decreasing of the electric current in a
loop, like when we disconnect a wire from a primary source of
electricity.



Decreasing electric current in a wire loop causes proportional decrease
of a magnetic field intensity at any point around a wire.

This, in turn, causes proportional decrease of magnetic flux flowing through a wire loop with rate of change (first derivative of flux by time) negative.

Negative rate of change of magnetic flux causes the generation of secondary (induced) EMF in a wire, that results in a secondary electric current, that interferes with the decreasing primary one.



Let's analyze the direction of this secondary electric current.

Above, when the primary electric current was increasing and the rate of change of magnetic flux was positive,
we concluded that induced EMF causes the secondary electric current to
go in the opposite to primary direction to preserve the Law of Energy
Conservation.



In a case of decreasing primary electric current we see that the
magnetic flux flowing through a wire loop is decreasing, it's rate of
change (first derivative) is negative, so the secondary (induced) EMF must be directed opposite to the one generated by an increasing primary electric current.



Therefore, if in the case of increasing primary electric current the secondary EMF "worked" against the electric current,
generating a secondary current in a direction opposite to an increasing
primary one, in the case of decreasing primary current the secondary (induced) EMF generates the secondary current in the same direction as a decreasing primary current, "helping" the decreasing primary current, prolonging the decrease.



So, in both cases the secondary (induced) EMF works opposite to a rate of change of the magnetic flux going through a wire loop.

When the magnetic flux increases, the induced EMF generates an
electric current in the opposite direction to a primary one, thus
slowing the rate of increase of the electric current and, therefore,
against the rate of increase of the magnetic flux.

When the magnetic flux decreases, the induced EMF generates an
electric current in the same direction as a primary one, thus slowing
the decrease of the electric current and, therefore, against the rate of
decrease of the magnetic flux.



The above considerations are the primary reason why the Faraday's Law is expressed as
U = −dΦ/dt

where the minus sign signifies that the induced EMF works against the rate of change of a magnetic flux.