P EM 3. Time Varying Electromagnetic Fields

8/7/2019 P EM 3. Time Varying Electromagnetic Fields

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The basic relationships of electrostatics and the magnetostatics field were obtained

in the previous chapters, and we are now ready to discuss time-varying fields. The discussion

will be short, for vector analysis and vector calculus should now be more familiar tools; some

of the relationships are unchanged, and most of the relationships are changed only slightly.

Two new concepts will be introduced: the electric field produced by a changing

magnetic field and the magnetic field produced by a changing electric filed. The first of these

concepts resulted from experimental research by Michael Faraday, and the second from

theatrical efforts of James Clerk Maxwell.

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In 1831 Michael Faraday performed experiments to check whether current is produced

in a closed wire loop placed near a magnet, in analogy to dc currents producing magnetic fields.

His experiment showed that this could not be done, but Faraday realized that a time

varying current in the loop was obtained while the magnet was being moved toward it or away

from it.

The law he formulated is known as Faradays law of electromagnetic induction. It is

perhaps the most important law of electromagnetism.

Without it there would be no electricity from rotating generators, no telephone, no

radio and television, no magnetic memories, to mention but a few applications.

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The phenomenon of electromagnetic induction has a simple physical interpretation.

Two charged particles (charges) at rest act on each other with a force given byCoulomb

law.

Two charges moving with uniform velocities act on each other with an additional

force, the magnetic force. If a particle is accelerated, there is another additional force that i

exerts on other charged particles, stationary or moving.

As in the case of the magnetic force, if only a pair of charges is considered, this

additional force is much smaller than Coulombs force.

However, time-varying currents in conductors involve a vast number of accelerated

charges, and produce effects significant enough to be easily measurable.

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The induced electric field and electromagnetic induction have immense practical

consequences. Some examples include:

1. The electric field of electromagnetic waves (e.g., radio waves or light) isbasically the induced electric field;

2. In electrical transformers, the induced electric field is responsible forobtaining higher or lower voltage than the input voltage;

3. The skin effect in conductors with ac currents is due to induced electric field;4. Electromagnetic induction is also the cause of magnetic coupling that may

result in undesired interference between wires (or metal traces) in any

system with time-varying current, an effect that increases with frequency.

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Faraday's original experiments consisted of a conducting loop through which he could impose

a dc current via a switch. Another short circuited loop with no source attached was nearby, as

shown in Figure 3.1. When a dc current flowed in loop 1, no current flowed in loop 2. However,

when the voltage was first applied to loop 1 by closing the switch, a transient current flowed in the

opposite direction in loop 2.

Figure 3. 1: Faraday's

experiments showed that a

time varying magnetic flux

through a closed conducting

loop induced a current in the

direction so as to keep the

flux through the loopconstant.

When the switch was later opened, another transient current flowed in loop 2, this time in the

same direction as the original current in loop 1. Currents are induced in loop 2 whenever a time

varying magnetic flux due to loop 1 passes through it. 7


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VEMF = .= =

.

(1))

The variation of flux with time as in eq. (1) may be caused in three ways:

1. By having a stationary loop in a time-varying B field.2. 2. By having a time-varying loop area in a static B field3. 3. By having a time-varying loop area in a time-varying B field.Each of these will be considered separately.


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1. STATIONARY LOOP IN TIME-VARYING B FIELD (TRANSFORMER

EMF)

This is the case is where a stationary conducting loop is in a time varying magnetic B

field. Equation (1) becomes

VEMF = . = . (2)This emf induced by the time-varying current (producing the time-varying B field) in a

stationary loop is often referred to as transformer emf in power analysis since it is due to

transformer action. By applying Stokes's theorem to the middle term in eq. (2), we obtain

. = dS = . Which is equivalent to:

+ . = 0


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1. STATIONARY LOOP IN TIME-VARYING B FIELD (TRANSFORMER

EMF)

Since this last relation is true for any surface, the integrand itself must be zero, which

yields Faraday's law of induction in differential form as

= This is one of the Maxwell's equations for time-varying fields. It shows that the time

varying E field is not conservative ( 0). This does not imply that the principles of energyconservation are violated. The work done in taking a charge about a closed path in a time-

varying electric field, for example, is due to the energy from the time-varying magnetic field.


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2. MOVING LOOP IN STATIC BFIELD (MOTIONAL EMF)When a conducting loop is moving in a static B field, an emf is induced in the loop. We

recall that the force on a charge moving with uniform velocity u in a magnetic field B is:

= We define the motional electric field as

= = If we consider a conducting loop, moving with uniform velocity u as consisting of a large

number of free electrons, the emf induced in the loop is

VEMF = . = . This type of emf is called motional emforflux-cutting emfbecause it is due to motional

action. It is the kind of emf found in electrical machines such as motors, generators, and

alternators.


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3. MOVING LOOP IN STATIC BFIELD (MOTIONAL EMF)

By applying Stokess theorem:

. d = . Or

=


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A conducting bar can slide freely over two conducting rails as shown in Figure Calculate theinduced voltage in the bar


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For static EM fields, we recall that

= But the divergence of the curl of any vector field is identically zero. Hence,

. = . = 0However, the continuity equation of current is given by

. = Thus,

= + d where d is to be determined and defined. Again, the divergence of the curl of anyvector is zero. Hence:

. = . + . d = 0


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Thus,

. d = . = = . = .

Or

d = Finally we can write

= + This is Maxwell's equation (based on Ampere's circuit law) for a time-varying field. The

term d = is known as displacement current densityand J is the conduction current density( = ).


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A parallel-plate capacitor with plate area of5 cm2 and plate separation of 3 mm has

a voltage 50sin103V applied to its plates. Calculate the displacement currentassuming = 20.


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The laws of electromagnetism that Maxwell put together in the form of four

equations were presented in previous chapter for static conditions. The more generalizedforms of these equations are those for time-varying conditions summarized in this section.

Differential Form Integral Form Remarks

. = . = Gausss law

. = 0 .

= Nonexistence of isolatedmagnetic charge

=

.

=

.

Faradays law

= + . = + . Ampers circuit law


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For a field to be "qualified" as an electromagnetic field, it must satisfy all fou

Maxwell's equations. The importance of Maxwell's equations cannot be overemphasize

because they summarize all known laws of electromagnetism.


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Just as the potential energy in an electrostatic field was derived as

= 12. =1

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Thus the energy in a magnetostatic field in a linear medium is

= 12. = 1

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P EM 3. Time Varying Electromagnetic Fields

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