Faraday Law

Electrostatic or magnetostatic fields are not coupled together.
(one can exist without the other.)
Electrostatic fields ! steady electric fields produced by stationary

electric charge.
Magnetostatic fields ! steady magnetic fields produced by steady

(DC) currents or stationary magnetic materials.
Dynamic electric and magnetic fields are coupled together.

(one field cannot exist without the other for time-varying fields.)
Electromagnetic fields ! produced by time-varying currents or

charges, or static sources in motion.
Maxwell’s equations ! four laws which govern the behavior of all
electromagnetic fields [Gauss’s law, Faraday’s law, Gauss’s law for
magnetic fields, Ampere’s law].
The static versions of Faraday’s law and Ampere’s law must be modified
to account for dynamic fields.
Complete Form of Faraday’s Law (Dynamic Fields)
The complete form of Faraday’s law, valid for both static and
dynamic fields, is defined in terms of a quantity known as the
electromotive force (emf). In an electric circuit, the emf is the force which
sets the charge in motion (forcing function for the current).
Example (emf in a battery/resistor circuit)

E = Ee + Ef (total electric field)
In general, the integral of the total electric field around a closed

circuit yields the total emf in the integration path.
Assumptions for the battery circuit:

(1) The emf electric field is confined to the battery.
(Ef = 0 outside the battery.)
(2) Connecting wires are perfect conductors.
(E = Ee = 0 inside the wires.)
The emf voltage in terms of the electric field components is
The resistor voltage in terms of the electric field components may be

determined using the conservative property of Ee.
Potential Difference Definition
General Equations for EMF and Potential Difference

Given the definition of electromotive force, we may now write the
dynamic form of Faraday’s law.
Faraday’s Law ! a time-changing magnetic flux through a closed

circuit induces an emf in the circuit (closed circuit ! induced current, open
circuit ! induced voltage).
The emf is an equal and opposite reaction to

the flux change (Lenz’s law)
The unit normal associated with

the differential surface ds is
related to the unit vector of the
differential length dl by the right
hand rule.
Note that when static fields are assumed, the time derivative on the right
hand side of the dynamic (complete) version of Faraday’s law goes to zero
and the equation reduces to the electrostatic form.
Example (Faraday’s law induction, wire loop in a time changing B)
For the closed loop, the flux produced by the induced current opposes
the change in B.
For the open-circuited loop, the polarity of the induced emf is defined
by the emf line integral.
Induction Types
1. Stationary circuit / time-varying B (transformer induction).

2. Moving circuit / static B (motional induction).
3. Moving circuit / time-varying B
(general case, transformer and motional induction).
Example (transformer induction ! AM antenna)
A circular wire loop of radius a = 0.4m lies in the x-y plane with its
axis along the z-axis. The vector magnetic field over the surface of the
loop is H = Ho cos(Tt)az where Ho = 200 :A/m and f = 1 MHz. Determine
the emf induced in the loop.
Since the loop is stationary, ds is

not time-dependent so that the
derivative with respect to time
can be brought inside the
integral.
The time derivative is written as a partial derivative since the magnetic flux
density is, in general, a function of both time and space. The polarity of
the induced emf is assigned when the direction of ds is chosen. If we
choose ds = az ds (then dl = aN dl for the line integral of E), the polarity of
the induced emf is that shown above. For this problem, both B [B = :o H]
and ds are az-directed so that the dot product in the transformer induction
integral is one.
Since the partial derivative of H with respect to time is independent of
position, it can be brought outside the integral. The resulting integral of
ds over the surface S yields the area of the loop so that
where A is the area of the loop (A = Ba2 ).
A typical AM antenna achieves a larger induced emf by employing

multiple turns of wire around a ferrite core.
Example (motional induction ! moving conductor / static B)
A particle of charge Q moving with velocity u in a uniform B

experiences a force given by
From Faraday’s law,
Choosing dl counterclockwise assigns the induced emf polarity as shown

above. On the moving conductor, dl = dy ay.
Note that a uniform velocity yields a DC voltage. An oscillatory

motion (back and forth) could be used to produce a sinusoidal voltage.
Example (General induction ! moving conductor / time-varying B)
Using the same geometry as the last example, assume that the
magnetic flux density is B = Bo cos Tt (!az).
Choose dl counterclockwise Y ds out

dl = dyay (on moving conductor) ds = dxdyaz
If we let x = 0 at t = 0 be our reference, then x = uo t and

Summary of Induction Formulas
Et = transformer emf electric field
Em = motional emf electric field
E = total emf electric field

Faraday’s Law (Differential Form)
The differential form of Faraday’s law can be found by applying

Stoke’s theorem to the integral form.
By applying Stoke’s theorem, the line integrals in the various forms of

Faraday’s law can be transformed into surface integrals. The integrands
of the surface integrals can then be equated to find the corresponding
differential form of the equation.
Displacement Current
(Maxwell’s contribution to Maxwell’s equations)
The concept of displacement current can be illustrated by considering

the currents in a simple parallel RC network (assume ideal circuit
elements, for simplicity).
iR(t) ! conduction
current
iC(t) ! displacement
current
From circuit theory Y

In the resistor, the conduction current model is valid (JR = FR ER ). The
ideal resistor electric field (ER) and current density (JR) are assumed to be
uniform throughout the volume of the resistor.
The conduction current model does not characterize the capacitor
current. The ideal capacitor is characterized by large, closely-spaced plates
separated by a perfect insulator (FC = 0) so that no charge actually passes
throught the dielectric [JC (t) = FC EC (t)]. The capacitor current measured
in the connecting wires of the capacitor is caused by the charging and
discharging the capacitor plates. Let Q(t) be the total capacitor charge on
the positive plate.
Based on these results, the static version of Ampere’s law must be

modified for dynamic fields to include conduction current AND
displacement current. Note that displacement current does not exist under
static conditions. The general form for current density in the dynamic field
problem is
displacement conduction convection

+
current current current
Complete Form of Ampere’s Law (Dynamic Fields)
Given the definition of displacement current, the complete form of

Ampere’s law for dynamic fields can be written.
The corresponding differential form of Ampere’s law is found using

Stoke’s theorem.
Since the two surface integrals above are valid for any surface S, we may
equate the integrands.
Example (Ampere’s law, non-ideal capacitor)
The previously considered parallel RC network represents the

equivalent circuit of a parallel plate capacitor with an imperfect insulating
material between the capacitor plates (finite conductivity).
Capacitor with imperfect

insulating material
(assume E, J are uniform)
Equivalent circuit
C ! models charge storage (displacement current)
R ! models leakage current (conduction current)
Let the applied voltage be a sinusoid. Y V(t) = Vo sin Tt
The resulting electric field in the capacitor is given by

Note that:
1. The peak conduction current density is independent of
frequency.
2. The peak displacement current density is directly proportional
to frequency.
3. The displacement current density leads the conduction current
density by 90o.
Since typical material permittivities are in the 1-100 pF/m range, the
displacement current density is typically negligible at low frequencies in
comparison to the conduction current density (especially in good
conductors). At high frequencies, the displacement current density
becomes more significant and can even dominate the conduction current
density in good insulators.
Maxwell’s Equations
(Dynamic fields)
In addition to his contribution of displacement current, Maxwell

brought together the four basic laws governing electric and magnetic fields
into one set of four equations which, as a set, completely describe the
behavior of any electromagnetic field. All of the vector field, flux, current
and charge terms in Maxwell’s equations are, in general, functions of both
time and space [e.g., E(x,y,z,t)]. The form of these quantities is referred
to as the instantaneous form (we can describe the fields at any point in
time and space). The instantaneous form of Maxwell’s equations may be
used to analyze electromagnetic fields with any arbitrary time-variation.
Maxwell’s Equations [instantaneous, differential form]
Maxwell’s Equations [instantaneous, integral form]

Constitutive Relations (linear, homogeneous, isotropic media)
Boundary Conditions
Note that the unit normal n points into region 2.

Faraday Law

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Electrostatic or magnetostatic fields are not coupled together.

(one can exist without the other.)

Electrostatic fields ! steady electric fields produced by stationary

Magnetostatic fields ! steady magnetic fields produced by steady

Dynamic electric and magnetic fields are coupled together.

Electromagnetic fields ! produced by time-varying currents or

Complete Form of Faraday’s Law (Dynamic Fields)

Example (emf in a battery/resistor circuit)

In general, the integral of the total electric field around a closed

Assumptions for the battery circuit:

The emf voltage in terms of the electric field components is

The resistor voltage in terms of the electric field components may be

General Equations for EMF and Potential Difference

Faraday’s Law ! a time-changing magnetic flux through a closed

The emf is an equal and opposite reaction to

The unit normal associated with

1. Stationary circuit / time-varying B (transformer induction).

Example (transformer induction ! AM antenna)

Since the loop is stationary, ds is

where A is the area of the loop (A = Ba2 ).

A typical AM antenna achieves a larger induced emf by employing

A particle of charge Q moving with velocity u in a uniform B

From Faraday’s law,

Choosing dl counterclockwise assigns the induced emf polarity as shown

Note that a uniform velocity yields a DC voltage. An oscillatory

Choose dl counterclockwise Y ds out

If we let x = 0 at t = 0 be our reference, then x = uo t and

Et = transformer emf electric field

Em = motional emf electric field

E = total emf electric field

The differential form of Faraday’s law can be found by applying

By applying Stoke’s theorem, the line integrals in the various forms of

The concept of displacement current can be illustrated by considering

From circuit theory Y

Based on these results, the static version of Ampere’s law must be

displacement conduction convection

Given the definition of displacement current, the complete form of

The corresponding differential form of Ampere’s law is found using

The previously considered parallel RC network represents the

Capacitor with imperfect

C ! models charge storage (displacement current)

R ! models leakage current (conduction current)

Let the applied voltage be a sinusoid. Y V(t) = Vo sin Tt

The resulting electric field in the capacitor is given by

In addition to his contribution of displacement current, Maxwell

Maxwell’s Equations [instantaneous, differential form]

Maxwell’s Equations [instantaneous, integral form]

Note that the unit normal n points into region 2.

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