Electromagnetic Waves, Antennas, and RF: Maxwell’s Equations as a Complete System - Exercises and Complete Worked Solutions

This companion article extends EM15 from recognition of Maxwell’s four equations to active use of the complete system. The problems emphasize geometric interpretation, dimensional checks, reconstruction of sources from fields, equivalence of integral and differential forms, constraint propagation, cylindrical induction problems, and a first plane-wave calculation that prepares the transition to the electromagnetic wave equation.

The complete differential system in vacuum with charge density ρ and current density J is

(1)

(2)

(3)

and

(4)

The exercises below should be worked by treating Maxwell’s Equations as one coupled field system rather than four unrelated formulas [1, 2, 3, 4, 5].

How to use this problem set

Attempt all exercises in Part I before consulting Part II. For every problem, first identify which Maxwell equation is relevant and whether the geometry suggests a divergence theorem, Stokes theorem, local differential calculation, or symmetry argument. Keep the following distinctions explicit:

• divergence versus curl;
• closed surfaces versus closed contours;
• charge density ρ versus current density J;
• conduction current versus displacement-current contribution;
• field constraints versus field-evolution equations;
• source-free vacuum versus field-free space.

Part I: Exercises

Exercise 1: structural map of Maxwell’s equations

For each of the four differential Maxwell equations:

1. classify it as a divergence law or curl law;
2. identify the corresponding integral theorem used to obtain the integral form;
3. state the geometry of the integral form: closed surface or closed contour plus spanning surface;
4. identify the physical source or dynamical quantity appearing on the right-hand side.

Then explain why the two divergence laws are naturally interpreted as constraints while the two curl laws contain the explicit time coupling that drives field evolution.

Exercise 2: dimensional consistency of the complete system

Verify the dimensions of each Maxwell equation in SI units.

Show explicitly that

(5)

(6)

and

(7)

Use

(8)

Explain why dimensional agreement is a useful but not sufficient test of a field equation.

Exercise 3: an electric field with divergence but no curl

Consider

(9)

where α, β, and γ are constants.

Find:

1. ∇⋅ E;
2. ∇× E;
3. the charge density ρ required by Gauss’s Law;
4. the condition on α, β, and γ for the region to be charge-free.

If the field is static, state what Faraday’s law implies about ∂B∕∂t.

Exercise 4: a magnetic field with curl but zero divergence

Consider the static magnetic field

(10)

where b has units of tesla per meter.

Find:

1. ∇⋅ B;
2. ∇× B;
3. the current density J required by the static Ampere law;
4. the direction of that current density.

Explain how this example distinguishes zero magnetic divergence from zero magnetic field.

Exercise 5: derive integral forms from local equations

Carry out both derivations in detail.

1. Starting from

   (11)

   integrate over an arbitrary volume V and derive Gauss’s integral law.

2. Starting from

   (12)

   integrate over a fixed surface S and derive Faraday’s integral law.

State exactly where the divergence theorem and Stokes’ theorem are used, and state why a fixed surface permits the time derivative to be moved outside the surface integral in part (b).

Exercise 6: cylindrical Faraday induction

A spatially uniform magnetic field points in the +z direction inside a circular cylindrical region of radius R and is negligible outside that region. The field changes with time:

(13)

Assume cylindrical symmetry.

1. Use Faraday’s law to derive the induced azimuthal electric field E_ϕ(r,t) for r < R.
2. Derive E_ϕ(r,t) for r > R.
3. Show that the two expressions agree at r = R.
4. Let

   (14)

   with B₀ = 3.0 mT, f = 60 Hz, and R = 4.0 cm. Find the maximum magnitude of E_ϕ at r = 3.0 cm and at r = 10.0 cm.

Exercise 7: cylindrical Ampere–Maxwell induction

A spatially uniform electric field points in the +z direction inside a circular region of radius R and is negligible outside that region:

(15)

Assume there is no conduction current through the spanning surface.

1. Use the Ampere–Maxwell law to derive B_ϕ(r,t) for r < R.
2. Derive B_ϕ(r,t) for r > R.
3. Show that the two expressions agree at r = R.
4. For R = 5.0 cm and

   (16)

   find the magnetic-field magnitude at r = 2.0 cm and r = 10.0 cm.

Exercise 8: reconstruct the sources from specified fields

Suppose the fields in a region are

(17)

and

(18)

where a and u are constants with appropriate units.

Use the complete Maxwell system to determine:

1. the charge density ρ;
2. whether Faraday’s law is satisfied;
3. the current density J required by the Ampere–Maxwell equation;
4. whether the resulting ρ and J satisfy the continuity equation.

This exercise should be treated as a source-reconstruction problem: the fields are given first, and the compatible sources must be inferred.

Exercise 9: why the Ampere–Maxwell integral is surface independent

Let S₁ and S₂ be two different surfaces having the same boundary contour C. Define the effective current flux through a surface by

(19)

Show that

(20)

provided charge continuity and Gauss’s electric law hold.

Your proof should:

1. combine S₁ and S₂ into a closed surface;
2. use the divergence theorem;
3. evaluate

   (21)

4. explain physically why the displacement-current term is necessary.

Exercise 10: propagation of Maxwell’s divergence constraints

The magnetic divergence law and Gauss’s electric law can be written as constraints on the instantaneous fields:

(22)

(23)

1. Take the divergence of Faraday’s law and show that

   (24)

2. Take the divergence of the Ampere–Maxwell law, use charge continuity, and show that

   (25)

3. Explain what these results mean for initial data in an electromagnetic field calculation.

Exercise 11: plane-wave precursor directly from the curl equations

In source-free vacuum, consider fields of the form

(26)

(27)

Do not use the electromagnetic wave equation. Use only the two source-free curl equations.

1. Apply Faraday’s law and derive a relation among E₀, B₀, k, and ω.
2. Apply the Ampere–Maxwell law and derive a second relation.
3. Combine the two relations to show that

   (28)

4. Show that

   (29)

5. For f = 1.57542 GHz, compute the vacuum wavelength λ.

Exercise 12: reject a longitudinal vacuum-wave candidate

Consider the proposed source-free vacuum field

(30)

1. Compute ∇⋅ E.
2. Determine whether Gauss’s electric law is satisfied in source-free vacuum for nonzero k and E₀.
3. Compute ∇× E and discuss Faraday’s law.
4. Explain why satisfying one Maxwell equation is not enough; a candidate field must satisfy the complete system.

Use this result to motivate why free-space electromagnetic plane waves are transverse rather than longitudinal.

Exercise 13: recover electrostatics and magnetostatics as limits

Assume all fields are time independent:

(31)

Reduce the complete Maxwell system to its static form.

Then answer:

1. what does ∇× E = 0 imply locally about an electrostatic potential V ?
2. what does the Ampere–Maxwell equation reduce to?
3. which two divergence equations remain unchanged?
4. why is magnetostatics not obtained by deleting Gauss’s magnetic law?

Exercise 14: source-free does not mean field-free

Determine whether each of the following is a valid source-free, time-independent vacuum solution of Maxwell’s equations:

1. E = E₀x, B = 0;
2. E = 0, B = B₀z;
3. E = E₀xx, B = 0;
4. E = 0, B = b(−yx + xy).

For each case, evaluate enough divergence and curl quantities to justify the answer. State the source that would be required when a field is not source-free.

Exercise 15: Julia finite-difference check of divergence and curl

Use central finite differences to numerically verify Exercises 3 and 4 at a point away from any boundary.

Take

(32)

and

(33)

Use the point

(34)

and finite-difference step

(35)

Write a short Julia program that estimates:

1. ∇⋅ E;
2. ∇× E;
3. ∇⋅ B;
4. ∇× B.

Compare the numerical values with the exact analytic results.

Part II: Complete Worked Solutions

Solution 1: structural map of Maxwell’s equations

Gauss’s electric law is

(36)

It is a divergence law. Integrating over a volume and applying the divergence theorem produces a closed-surface flux equation. Its source is electric charge density ρ.

Gauss’s magnetic law is

(37)

It is also a divergence law, and the divergence theorem gives the closed-surface result

(38)

Its zero right-hand side expresses the absence of magnetic monopole source density in classical Maxwell theory.

Faraday’s law is

(39)

It is a curl law. Stokes’ theorem converts its surface integral into circulation around the boundary contour. The dynamical quantity on the right is the local time rate of change of B.

The Ampere–Maxwell law is

(40)

It is also a curl law. Its integral form uses a closed contour and spanning surface. Magnetic circulation is sourced by conduction current density and by changing electric field.

The divergence equations contain no explicit first-order time derivatives of the fields. They constrain allowable field configurations at each instant. The curl equations explicitly connect time derivatives of one field to spatial derivatives of the other, so they form the core evolution coupling.

Solution 2: dimensional consistency of the complete system

Because a spatial derivative contributes one inverse meter,

(41)

Also,

(42)

so

	=	(43)
	= V/m2.	(44)

Thus Gauss’s electric law is dimensionally consistent.

For Faraday’s law,

(45)

Using

(46)

we obtain

(47)

For the magnetic curl equation,

(48)

Since

(49)

we have

(50)

Finally,

(51)

so

	=	(52)
	=	(53)
	= T/m.	(54)

Dimensional agreement is necessary because terms added or equated must share dimensions. It is not sufficient because an incorrect equation can still be constructed with the correct units.

Solution 3: an electric field with divergence but no curl

The field is

(55)

Its divergence is

∇⋅ E	= + +	(56)
	= α + β + γ.	(57)

Therefore,

(58)

Every cross derivative vanishes, so

(59)

The charge-free condition is therefore

(60)

If the field is static, Faraday’s law gives

(61)

so

(62)

Solution 4: a magnetic field with curl but zero divergence

The field components are

(63)

The divergence is

∇⋅ B	= + +	(64)
	= 0.	(65)

Thus Gauss’s magnetic law is satisfied.

The curl has only a z component:

(∇× B)z	= −	(66)
	= b − (−b)	(67)
	= 2b.	(68)

Hence

(69)

For a static field, Ampere–Maxwell reduces to

(70)

Therefore,

(71)

The current density points in the +z direction for positive b. Zero magnetic divergence means no net magnetic source or sink; it does not require the magnetic field or its curl to vanish.

Solution 5: derive integral forms from local equations

For Gauss’s electric law, integrate over an arbitrary volume V :

(72)

Apply the divergence theorem:

(73)

Since

(74)

we obtain

(75)

For Faraday’s law, integrate over a fixed surface S:

(76)

Apply Stokes’ theorem to the left side:

(77)

Because the surface is fixed in space and does not change shape,

(78)

Therefore,

(79)

For a moving or deforming surface, additional transport terms generally appear; the simple interchange above is tied to a fixed integration surface.

Solution 6: cylindrical Faraday induction

By symmetry the induced electric field is azimuthal and has constant magnitude around a circular contour of radius r.

For r < R, Faraday’s law gives

(80)

Thus

(81)

For r > R, only the region of radius R contributes magnetic flux:

(82)

so

(83)

At r = R, either expression gives

(84)

so the idealized solution is continuous.

For

(85)

we have

(86)

With f = 60 Hz and B₀ = 3.0 × 10⁻³ T,

(87)

At r = 3.0 cm < R,

|Eϕ|max	= (1.131)	(88)
	= 1.70 × 10−2 V/m.	(89)

Thus

(90)

At r = 10.0 cm > R,

|Eϕ|max	= (1.131)	(91)
	= 9.05 × 10−3 V/m.	(92)

Therefore,

(93)

Inside the changing-flux region the induced field grows linearly with r; outside it falls as 1∕r because the enclosed changing flux has saturated at πR²B.

Solution 7: cylindrical Ampere–Maxwell induction

With no enclosed conduction current, the integral Ampere–Maxwell law is

(94)

By symmetry, B = B_ϕϕ.

For r < R,

(95)

Hence

(96)

For r > R,

(97)

so

(98)

At r = R, both expressions give

(99)

Using

(100)

at r = 0.020 m,

Bϕ	= (0.020)(2.0 × 1010)	(101)
	= 2.23 × 10−9 T.	(102)

Thus

(103)

At r = 0.100 m,

Bϕ	= (2.0 × 1010)	(104)
	= 2.78 × 10−9 T.	(105)

Therefore,

(106)

This is the electric-field analogue of Exercise 6: changing electric flux generates circulating magnetic field.

Solution 8: reconstruct the sources from specified fields

The electric field is

(107)

Its divergence is

(108)

Gauss’s law therefore gives

(109)

The curl of this electric field is zero because it has only an x component and that component depends only on x and t:

(110)

Since B = 0, also

(111)

Therefore Faraday’s law is satisfied.

Now use the Ampere–Maxwell equation. Since B = 0,

(112)

But

(113)

Hence

(114)

Finally,

(115)

and the reconstructed current is spatially uniform, so

(116)

Thus

(117)

The sources inferred independently from Maxwell’s equations are automatically compatible with charge conservation.

Solution 9: why the Ampere–Maxwell integral is surface independent

Reverse the orientation of S₂ and join it to S₁. The two surfaces form a closed surface S_c.

The difference between the effective currents is

(118)

Apply the divergence theorem:

(119)

Evaluate the divergence:

∇⋅	= ∇⋅ J + 𝜖0(∇⋅ E)	(120)
	= ∇⋅ J + 𝜖0	(121)
	= ∇⋅ J + .	(122)

Charge continuity states

(123)

Therefore the integrand is zero everywhere, giving

(124)

Without the displacement-current term, the divergence would be simply ∇⋅ J = −∂ρ∕∂t, which need not vanish during charge accumulation. The magnetic circulation around one contour would then depend on which spanning surface was chosen. Maxwell’s correction removes that inconsistency.

Solution 10: propagation of Maxwell’s divergence constraints

Take the divergence of Faraday’s law:

(125)

The divergence of a curl is identically zero, so

(126)

Thus if ∇⋅ B = 0 at the initial time, ideal Maxwell evolution preserves it.

Now take the divergence of the Ampere–Maxwell law:

(127)

Divide by μ₀𝜖₀:

(128)

From charge continuity,

(129)

Therefore,

(130)

Rearranging,

(131)

The implication is important for analytical and numerical field evolution: initial fields must satisfy the divergence constraints. The ideal continuous equations preserve those constraints thereafter when the sources satisfy continuity. Numerical discretization can introduce constraint error, so computational electromagnetics often monitors or actively controls discrete divergence errors.

Solution 11: plane-wave precursor directly from the curl equations

Let

(132)

The electric field has only an x component, so its curl has only a y component:

(∇× E)y	=	(133)
	= −kE0 sin ϕ.	(134)

The time derivative of the magnetic field is

(135)

Faraday’s law gives

(136)

For nontrivial fields,

(137)

Next compute the magnetic curl. Since only B_y is nonzero,

(∇× B)x	= −	(138)
	= kB0 sin ϕ.	(139)

Also,

(140)

The source-free Ampere–Maxwell equation gives

(141)

so

(142)

Use the first relation to write

(143)

Substitute into the second:

(144)

Cancel E₀ and multiply by ω:

(145)

Therefore,

(146)

From kE₀ = ωB₀,

(147)

so

(148)

The wavelength follows from

(149)

For f = 1.57542 × 10⁹ Hz,

λ	=	(150)
	≈ 0.1903 m.	(151)

Hence,

(152)

This result has been obtained directly from Maxwell’s curl equations, before deriving the second-order wave equation.

Solution 12: reject a longitudinal vacuum-wave candidate

The proposed electric field is

(153)

Its divergence is

∇⋅ E	=	(154)
	= −kE0 sin(kz − ωt).	(155)

In source-free vacuum, Gauss’s electric law requires

(156)

For nonzero k and nonzero E₀, the proposed field does not satisfy that condition for all space and time. It therefore cannot be a source-free vacuum plane wave.

Because the field has only a z component that depends only on z and t,

(157)

With B = 0, Faraday’s law alone is satisfied. This demonstrates why one cannot validate an electromagnetic field by checking only one of Maxwell’s equations. The entire coupled system must hold simultaneously.

For a propagating source-free plane wave, Gauss’s electric law rules out a varying electric component parallel to the propagation direction. This is the beginning of the transversality result developed more fully in the wave lessons.

Solution 13: recover electrostatics and magnetostatics as limits

Set all time derivatives to zero. Maxwell’s equations become

(158)

(159)

(160)

and

(161)

In a simply connected region, ∇× E = 0 permits a scalar potential representation

(162)

The magnetic curl equation reduces to the ordinary magnetostatic Ampere law. Both divergence laws remain unchanged because neither contained an explicit time derivative to begin with.

Gauss’s magnetic law must remain because magnetostatics still requires magnetic fields to have zero net flux through every closed surface. The static limit removes time coupling; it does not remove the geometric constraint on B.

Solution 14: source-free does not mean field-free

For case (a),

(163)

is spatially uniform and static. Therefore

(164)

With B = 0, all four source-free static Maxwell equations are satisfied. This is a valid local vacuum solution.

For case (b), a uniform static magnetic field also has

(165)

Thus it too is a valid source-free local vacuum solution.

For case (c),

(166)

has

(167)

Therefore it requires

(168)

so it is not source-free.

For case (d), the field from Exercise 4 has

(169)

but

(170)

Thus it requires

(171)

so it is not source-free.

The first two cases make the key point: ρ = 0 and J = 0 do not imply E = 0 and B = 0.

Solution 15: Julia finite-difference check of divergence and curl

For the specified constants, the exact electric-field results are

(172)

and

(173)

For the magnetic field,

(174)

and

(175)

One compact Julia implementation is

using LinearAlgebra

alpha, beta, gamma = 2.0, -1.0, 4.0
b = 3.0
h = 1.0e-5
r0 = [0.7, -0.4, 1.1]

E(r) = [alpha*r[1], beta*r[2], gamma*r[3]]
B(r) = [-b*r[2], b*r[1], 0.0]

\href{https://physicslibrary.org/encyclopedia/Bijective.html}{Function} partial(F, r, component, coordinate, h)
rp = copy(r)
rm = copy(r)
rp[coordinate] += h
rm[coordinate] -= h
return (F(rp)[component] - F(rm)[component])/(2h)
end

function divF(F, r, h)
return partial(F,r,1,1,h) +
partial(F,r,2,2,h) +
partial(F,r,3,3,h)
end

function curlF(F, r, h)
cx = partial(F,r,3,2,h) - partial(F,r,2,3,h)
cy = partial(F,r,1,3,h) - partial(F,r,3,1,h)
cz = partial(F,r,2,1,h) - partial(F,r,1,2,h)
return [cx, cy, cz]
end

println("div E = ", divF(E,r0,h))
println("curl E = ", curlF(E,r0,h))
println("div B = ", divF(B,r0,h))
println("curl B = ", curlF(B,r0,h))

The numerical output should be close to

div E = 5.0
curl E = [0.0, 0.0, 0.0]
div B = 0.0
curl B = [0.0, 0.0, 6.0]

up to floating-point and finite-difference roundoff. The exercise connects the geometric definitions of divergence and curl with the discrete derivative operations used in computational field solvers.

1 What EM15E1 adds to the series

EM15 assembled Maxwell’s four equations into a coherent field theory. EM15E1 pushes that synthesis further by requiring the equations to be used together.

The most important structural results are:

(176)

(177)

(178)

and, for the transverse plane-wave trial fields,

(179)

The next natural step is to take the curl of the curl equations and derive the second-order electromagnetic wave equations for E and B directly.

References