lOMoARcPSD|16248954
Chapter 9
Time-Var ying Fields and Maxwell’s Equat ions
Two concepts will be introduced:
1. The electric field produced by a changing magnetic field (F
law).
2. The magnetic field produced by a changing electric field (M
equation).
9.1FARADAY’S LAW
Faraday discovered that the induced electromotive force (em
closed circuit is equal to the time rate of change of the mag
linkage by the circuit. This is called Faraday's law,
Faraday's law links electric and magnetic fields. The m
indicates that the induced voltage acts to produce an oppo
and is known as Lenz’s law.
For an N-turn filamentary conductor,
where Φ is now interpreted as the flux passing through any
coincident paths. The emf is a scalar, and it is measured in volt
We define the emf as,
With time-varying fields, the line integral result an emf or
about a specific closed path. In electrostatics, the line integra
, lOMoARcPSD|16248954
value of E which is opposite to the positive direction about th
path.
TRANSFORMER AND MOTIONAL EMFs
The emf may result from:
A. Stationary Loop in Time-Varying B Field (transformer em
We first consider a stationary conducting loop in a time
magnetic B field as shown in figure 9.3.
The magnetic flux is the only time-varying quantity on the rig
eq. (4), and a partial derivative may be taken under the integra
This emf induced by the time-varying B field in a stationa
often referred to as transformer emf since it is due to tra
action.
Figure 9.3 Induced emf due to a stationary loop in a tim
B field.
To obtain the point form of this integral equation we apply
theorem to the closed line integral, so:
, lOMoARcPSD|16248954
This is one of Maxwell’s four equations as written in differ
point form. Remembering the definition of curl, we see
electric field has the special property of circulation; its lin
about a general closed path eq. (5) is not zero.
If B is not a function of time, eq. (6) evidently reduce
electrostatic equations.
Eq. (6) shows that the time varying E field is not con
. This does not imply that the principles o
conservation are violated. The work done in taking a charg
closed path in a time-varying electric field, for example, is d
energy from the time-varying magnetic field.
B. Moving Loop in Static B Field (Motional emf)
When a conducting loop is moving in a static B field, an emf i
in the loop.
The force on a charge Q moving at a velocity v in a magnetic f
or
This force per unit charge is called the motional electric field
Em,
The motional emf produced by the moving conductor is then
where the last integral may have a nonzero value only a
Chapter 9
Time-Var ying Fields and Maxwell’s Equat ions
Two concepts will be introduced:
1. The electric field produced by a changing magnetic field (F
law).
2. The magnetic field produced by a changing electric field (M
equation).
9.1FARADAY’S LAW
Faraday discovered that the induced electromotive force (em
closed circuit is equal to the time rate of change of the mag
linkage by the circuit. This is called Faraday's law,
Faraday's law links electric and magnetic fields. The m
indicates that the induced voltage acts to produce an oppo
and is known as Lenz’s law.
For an N-turn filamentary conductor,
where Φ is now interpreted as the flux passing through any
coincident paths. The emf is a scalar, and it is measured in volt
We define the emf as,
With time-varying fields, the line integral result an emf or
about a specific closed path. In electrostatics, the line integra
, lOMoARcPSD|16248954
value of E which is opposite to the positive direction about th
path.
TRANSFORMER AND MOTIONAL EMFs
The emf may result from:
A. Stationary Loop in Time-Varying B Field (transformer em
We first consider a stationary conducting loop in a time
magnetic B field as shown in figure 9.3.
The magnetic flux is the only time-varying quantity on the rig
eq. (4), and a partial derivative may be taken under the integra
This emf induced by the time-varying B field in a stationa
often referred to as transformer emf since it is due to tra
action.
Figure 9.3 Induced emf due to a stationary loop in a tim
B field.
To obtain the point form of this integral equation we apply
theorem to the closed line integral, so:
, lOMoARcPSD|16248954
This is one of Maxwell’s four equations as written in differ
point form. Remembering the definition of curl, we see
electric field has the special property of circulation; its lin
about a general closed path eq. (5) is not zero.
If B is not a function of time, eq. (6) evidently reduce
electrostatic equations.
Eq. (6) shows that the time varying E field is not con
. This does not imply that the principles o
conservation are violated. The work done in taking a charg
closed path in a time-varying electric field, for example, is d
energy from the time-varying magnetic field.
B. Moving Loop in Static B Field (Motional emf)
When a conducting loop is moving in a static B field, an emf i
in the loop.
The force on a charge Q moving at a velocity v in a magnetic f
or
This force per unit charge is called the motional electric field
Em,
The motional emf produced by the moving conductor is then
where the last integral may have a nonzero value only a
