Electromagnetic Waves, Antennas, and RF: Faraday’s Law and Electromagnetic Induction

EM05–EM12 developed charges, currents, electric flux, magnetic fields, magnetic forces, the Biot–Savart law, and Ampère’s law under predominantly static or steady-current conditions. EM13 introduces the first fundamental mechanism that couples the electric and magnetic fields dynamically:

(1)

Faraday’s law is one of the central experimental laws of Electromagnetism. In modern field language it states that a time-varying magnetic field is associated with a circulating electric field. This is a major conceptual step toward Maxwell’s equations, electromagnetic waves, radio antennas, and RF receivers [1, 2, 3, 5].

1 Magnetic flux

The magnetic flux through an oriented surface S is

(2)

The oriented area element is

(3)

where n is the chosen surface normal.

For a uniform magnetic field through a flat surface of area A,

(4)

where 𝜃 is the angle between B and the surface normal, not the surface itself.

The SI unit of magnetic flux is the weber:

(5)

2 Electromotive force

The term electromotive force, abbreviated emf, is historical. An emf is not a mechanical force. It is work done per unit charge around a circuit and therefore has units of volts.

For a stationary closed contour C, the electric contribution to emf is

(6)

If the electric field were purely electrostatic, this closed-loop integral would vanish. Electromagnetic induction introduces a different kind of electric field: one whose circulation around a closed path can be nonzero.

3 Faraday’s law for a stationary loop

For a fixed loop and fixed spanning surface, Faraday’s law states

(7)

For a coil of N identical tightly coupled turns,

(8)

The negative sign is Lenz’s law. It encodes the direction of the induced emf relative to the chosen loop orientation and magnetic-flux direction.

The magnetic flux can change because one or more of the following changes:

• the magnetic-field magnitude B;
• the area A of the loop;
• the orientation angle 𝜃;
• the spatial relationship between a nonuniform field and the loop.

4 Example 1: changing uniform magnetic field

A circular loop has radius

(9)

A uniform magnetic field normal to the loop increases from

(10)

to

(11)

in

(12)

The loop area is

(13)

Because the field is normal to the loop,

(14)

Thus,

|ℰ|	= (2.01 × 10−2)	(15)
	= 2.01 × 10−2 V.	(16)

Therefore,

(17)

The sign and current direction require an orientation convention and Lenz’s law.

5 Lenz’s law

Lenz’s law gives a physical interpretation of the minus sign in Faraday’s law:

(18)

This does not mean the induced magnetic field always points opposite to the external magnetic field. It opposes the change.

For example, if an external magnetic field into the page is increasing, an induced current produces a magnetic field out of the page. If the same external field into the page is decreasing, the induced response instead tends to maintain an into-page field.

6 Orientation conventions and the minus sign

The orientation of the loop and the orientation of the spanning surface are linked by the right-hand rule.

If the fingers curl in the positive traversal direction around C, the thumb gives the positive normal n. Magnetic flux is positive when B has a component along that normal.

With these linked orientations, the integral form of Faraday’s law for a stationary loop is

(19)

The minus sign is therefore not an arbitrary extra rule. It is part of the oriented relationship among the contour, surface normal, magnetic flux, and induced electric-field circulation.

7 Example 2: an N-turn coil

A coil has

(20)

turns and area per turn

(21)

The field is normal to the coil and changes at the rate

(22)

The emf magnitude is

|ℰ|	= NA	(23)
	= (200)(4.0 × 10−4)(0.50)	(24)
	= 4.0 × 10−2 V.	(25)

Hence,

(26)

8 Rotating-loop induction

Suppose a loop of area A rotates in a uniform magnetic field B with angular speed ω. If the loop normal makes angle

(27)

with the field, then

(28)

For an N-turn coil,

ℰ	= −N[BA cos(ωt)]	(29)
	= NBAω sin(ωt).	(30)

Therefore,

(31)

with peak emf

(32)

This is the idealized operating principle of an AC generator.

9 Example 3: rotating coil

Let

(33)

Then

ℰ0	= (50)(1.0 × 10−2)(0.20)(100)	(34)
	= 10 V.	(35)

Thus,

(36)

10 Motional emf

A changing magnetic flux can also occur because a Conductor moves through a magnetic field.

A charge moving with conductor velocity v experiences the magnetic Lorentz force

(37)

The magnetic force per unit charge is

(38)

For a moving conductor, the motional contribution to emf is

(39)

where the relevant velocity is the velocity of the conductor element.

For a straight rod of length L moving with speed v perpendicular to both the rod and a uniform magnetic field,

(40)

11 Example 4: sliding conducting rod

Let

(41)

Then

|ℰ|	= BLv	(42)
	= (0.40)(0.50)(3.0)	(43)
	= 0.60 V.	(44)

Therefore,

(45)

12 Transformer emf and motional emf are physically distinct

Two mechanisms can contribute to circuit emf:

1. Transformer emf: a time-varying magnetic field produces a circulating electric field even when the circuit is stationary.
2. Motional emf: magnetic Lorentz force acts on charges in a conductor moving through a magnetic field.

For a moving material circuit, the total Lorentz-force-per-charge circulation can be written schematically as

(46)

Under the usual circuit conditions, this total emf is consistent with the flux rule

(47)

where the derivative includes changes caused by the field, the circuit geometry, or both.

However, the local Maxwell–Faraday equation introduced below refers specifically to the electric field generated by a time-varying magnetic field. Keeping these mechanisms conceptually separate prevents later confusion.

13 A changing magnetic field produces a circulating electric field

The field form of Faraday’s law does not require a conducting wire. A changing magnetic field can produce an electric field in empty space.

For a stationary closed contour C,

(48)

The induced electric field is generally not electrostatic. Its field lines can form closed loops, and its closed-loop circulation can be nonzero.

14 Example 5: induced electric field inside a changing-field region

Suppose a spatially uniform magnetic field occupies a circular region and changes at the rate dB∕dt. Choose a circular observation contour of radius r lying completely inside that region.

By rotational symmetry, the induced electric field is tangent to the circle and has constant magnitude E(r). Therefore,

(49)

The enclosed magnetic flux is

(50)

Faraday’s law gives

(51)

Thus,

(52)

where the sign denotes the circulation direction relative to the chosen contour orientation.

The magnitude is

(53)

15 Example 6: induced electric field outside the changing-field region

Now suppose the changing magnetic field occupies only a circular region of radius R, while the observation contour has radius r > R.

Only the field-containing area contributes to the magnetic flux:

(54)

Therefore,

(55)

so

(56)

This field decreases as 1∕r outside the changing-flux region.

16 From integral Faraday law to the differential form

For a fixed contour and fixed spanning surface, Stokes’ theorem gives

(57)

Faraday’s law gives

(58)

For a fixed surface, the time derivative may be brought inside the surface integral:

(59)

Therefore,

(60)

For arbitrary surfaces,

(61)

This is the Maxwell–Faraday equation.

It connects directly back to the curl operator introduced in EM03. Curl is no longer merely a mathematical measure of local circulation: in electromagnetism, the curl of the electric field is tied directly to the local time variation of the magnetic field.

17 Example 7: interpreting the differential equation

Suppose at some point in space

(62)

Then Maxwell–Faraday gives

(63)

Thus the local electric-field circulation has negative z orientation. Using the right-hand rule, that corresponds to clockwise local circulation when viewed from the +z side.

18 Why Lenz’s law is required by energy conservation

Suppose the induced current reinforced the change in magnetic flux rather than opposing it. An increasing flux would produce a current that created still more flux in the same direction, which would drive still more current. The system would amplify itself without an external energy source.

Lenz’s law prevents that unphysical behavior. Mechanical work or another energy source is required to change the flux against the induced response.

For the sliding rod, for example, the induced current experiences a magnetic force opposing the imposed motion. To maintain constant speed, an external agent must do mechanical work. That mechanical power can appear as electrical power and ultimately as resistive heating in the circuit.

19 A first bridge to antennas and radio waves

Faraday’s law is one of the equations that makes electromagnetic waves possible. A time-varying magnetic field produces a circulating electric field:

(64)

Later, Maxwell’s correction to Ampère’s law will show the complementary relationship: a time-varying electric field contributes to the curl of the magnetic field.

Together, these coupled curl equations allow electric and magnetic disturbances to propagate through space as electromagnetic waves.

The induction viewpoint is also directly relevant to receiving antennas. A time-varying electromagnetic field can induce voltage and current in a conducting structure. The exact RF antenna problem requires the full Maxwell theory developed later, but Faraday’s law provides one of the fundamental pieces.

20 Common mistakes

• Mistake: measuring 𝜃 from the plane of the loop rather than from its normal. In Φ_B = BA cos 𝜃, 𝜃 is the angle between B and n.
• Mistake: saying that Lenz’s law opposes the magnetic field. It opposes the change in magnetic flux.
• Mistake: treating emf as a mechanical force. Emf has units of volts and represents work per unit charge.
• Mistake: assuming that a wire is required for an induced electric field. The Maxwell–Faraday field equation exists in space whether or not a conductor is present.
• Mistake: using BLv without checking that the rod, velocity, and magnetic field have the required perpendicular geometry.
• Mistake: mixing motional emf and transformer emf without recognizing their different local force mechanisms.
• Mistake: applying the simple stationary-contour integral form without considering whether the circuit itself is moving or deforming.

Part I: Exercises

All exercises are stated here before any worked solution. Attempt the complete set before proceeding to Part II.

Exercise 1: magnetic flux through a tilted loop

A uniform magnetic field has magnitude

(65)

A flat loop has area

(66)

and its area normal makes an angle

(67)

with the magnetic field. Find the magnetic flux.

Exercise 2: flux sign and orientation

A loop has area A and a magnetic field of magnitude B points exactly opposite to the chosen area normal.

Find the signed magnetic flux and explain what happens to the sign if the area-normal convention is reversed.

Exercise 3: induced emf from a changing field

A single-turn loop has area

(68)

A perpendicular magnetic field increases uniformly from 0.20 T to 0.50 T in 0.10 s. Find the emf magnitude.

Exercise 4: multiturn coil

A 150-turn coil has area per turn

(69)

A perpendicular magnetic field decreases at the constant rate

(70)

Find the induced emf magnitude.

Exercise 5: Lenz-law direction

A circular conducting loop is viewed face-on. An external magnetic field points out of the page and is increasing.

Determine:

1. the direction of the induced magnetic field;
2. whether the induced current is clockwise or counterclockwise.

Exercise 6: rotating-loop generator

A 40-turn coil of area

(71)

rotates at angular speed

(72)

in a uniform magnetic field

(73)

Find the peak induced emf.

Exercise 7: motional emf

A conducting rod of length

(74)

moves at speed

(75)

perpendicular to a uniform field

(76)

The rod is perpendicular to both v and B. Find the motional emf magnitude.

Exercise 8: magnetic force direction in a moving rod

A conducting rod moves in the +x direction through a magnetic field in the −z direction.

For a positive charge moving with the rod, find the direction of

(77)

Which end of a rod oriented along the y axis tends to become positively charged?

Exercise 9: induced electric field inside a changing-field region

A uniform magnetic field fills a circular region and changes at the rate

(78)

Find the magnitude of the induced electric field on a circular contour of radius

(79)

lying entirely inside the changing-field region.

Exercise 10: induced electric field outside a changing-field region

A uniform changing magnetic field occupies a circular region of radius

(80)

and changes at the rate

(81)

Find the induced electric-field magnitude at radius

(82)

Exercise 11: derive the differential Maxwell–Faraday equation

Starting from

(83)

for a fixed contour and surface, use Stokes’ theorem to derive

(84)

Exercise 12: energy and Lenz’s law

Explain why an induced current that reinforced the flux change that produced it would conflict with energy conservation. Use either a moving magnet and loop or a sliding conducting rod as your physical example.

Exercise 13: transformer emf versus motional emf

For each case below, identify the primary local mechanism producing emf:

1. a stationary wire loop in a time-varying magnetic field;
2. a conducting rod moving through a static magnetic field;
3. a moving loop in a magnetic field that is also changing with time.

State which cases can contain an electric-field contribution, a v × B contribution, or both.

Exercise 14: RF loop preview

A single-turn receiving loop has area

(85)

A locally uniform sinusoidal magnetic field normal to the loop is

(86)

with

(87)

Ignoring loading and radiation effects, find:

1. an expression for the induced emf ℰ(t);
2. the peak emf magnitude.

Part II: Complete Worked Solutions

Solution 1: magnetic flux through a tilted loop

For a uniform field through a flat area,

(88)

Substitute the values:

ΦB	= (0.35)(2.0 × 10−2) cos 60∘	(89)
	= (0.35)(2.0 × 10−2)(0.5)	(90)
	= 3.5 × 10−3 Wb.	(91)

Therefore,

(92)

Solution 2: flux sign and orientation

If B points opposite to the chosen normal, then

(93)

Therefore,

(94)

So

(95)

If the area-normal convention is reversed, the same physical field now points along the positive normal and the signed flux becomes

(96)

The physical configuration has not changed; only the orientation convention has changed.

Solution 3: induced emf from a changing field

The field change is

(97)

The emf magnitude is

|ℰ|	= A	(98)
	= (3.0 × 10−3)	(99)
	= 9.0 × 10−3 V.	(100)

Thus,

(101)

Solution 4: multiturn coil

For N turns,

(102)

Hence,

|ℰ|	= (150)(5.0 × 10−4)(0.80)	(103)
	= 6.0 × 10−2 V.	(104)

Therefore,

(105)

Solution 5: Lenz-law direction

The external field points out of the page and is increasing. The induced response opposes the increase, so the induced magnetic field must point into the page.

Thus,

(106)

Using the right-hand rule, a clockwise current produces a magnetic field into the page. Therefore,

(107)

Solution 6: rotating-loop generator

The peak emf is

(108)

Substitute:

ℰ0	= (40)(0.25)(8.0 × 10−3)(120)	(109)
	= 9.6 V.	(110)

Therefore,

(111)

Solution 7: motional emf

For the perpendicular sliding-rod geometry,

(112)

Hence,

|ℰ|	= (0.50)(0.30)(4.0)	(113)
	= 0.60 V.	(114)

Thus,

(115)

Solution 8: magnetic force direction in a moving rod

The rod velocity is

(116)

and the field is

(117)

Therefore,

v × B	∝x × (−z)	(118)
	= −x ×z	(119)
	= +y.	(120)

For a positive charge,

(121)

So the +y end of a rod oriented along the y axis tends to become positively charged.

Solution 9: induced electric field inside a changing-field region

Inside the changing-field region,

(122)

Substitute:

|E|	= (3.0)	(123)
	= 6.0 × 10−2 V/m.	(124)

Therefore,

(125)

Solution 10: induced electric field outside a changing-field region

For r > R,

(126)

Substitute:

|E|	= (2.5)	(127)
	= 4.0 × 10−2 V/m.	(128)

Hence,

(129)

Solution 11: derive the differential Maxwell–Faraday equation

Start with the stationary-contour integral form:

(130)

Apply Stokes’ theorem to the left-hand side:

(131)

For a fixed surface,

(132)

Therefore,

(133)

Because this holds for arbitrary fixed surfaces,

(134)

Solution 12: energy and Lenz’s law

Suppose an induced current reinforced the change in flux. If a magnet approached a conducting loop, the increasing flux would induce a current whose magnetic field increased the flux further. That stronger change would induce an even larger current, producing self-amplification without a corresponding energy input.

Instead, the induced current creates a magnetic effect that resists the imposed change. An external agent must therefore do work to continue moving the magnet or conductor. That work supplies the electrical energy that may later be dissipated as heat or stored in the electromagnetic system.

Thus Lenz’s law is consistent with energy conservation:

(135)

Solution 13: transformer emf versus motional emf

1. A stationary loop in a time-varying magnetic field has a transformer emf. The local mechanism is a nonconservative induced electric field, represented by E in the circulation integral.
2. A rod moving through a static magnetic field has a motional emf. The local mechanism is the magnetic Lorentz force per unit charge, v × B.
3. A moving loop in a field that also varies in time may have both contributions. The total emf may include both E and v × B terms.

Thus the general circuit-force-per-charge viewpoint is

(136)

Solution 14: RF loop preview

The magnetic field is

(137)

Because the field is normal to the loop,

(138)

Faraday’s law gives

ℰ(t)	= −	(139)
	= 2πfAB0 sin(2πft).	(140)

Therefore,

(141)

The peak magnitude is

(142)

Using

(143)

we obtain

ℰ0	= 2π(1.0 × 106)(1.0 × 10−4)(20 × 10−6)	(144)
	= 1.26 × 10−2 V.	(145)

Hence,

(146)

This simple result is only a preview. A practical RF loop antenna must also be treated using circuit loading, self-inductance, impedance, radiation, field polarization, and the full Maxwell equations.

21 What EM13 adds to the series

EM12 established the magnetostatic curl relation

(147)

EM13 adds the first explicitly time-dependent field coupling:

(148)

The next major step is to complete Ampère’s law with Maxwell’s displacement-current term. Once both curl equations contain time-dependent field terms, the mathematical structure needed for self-propagating electromagnetic waves is nearly complete.

References