AP Physics C - Maxwell’s Equations and Electromagnetic Waves

This post is meant for AP Physics C aspirants who are required to have some idea about Maxwell’s equations and their consequence.

Maxwell’s equations are basically the mathematical statements of

(i) Gauss’s law in electricity

(ii) Gauss’s law in magnetism

(iii) Faraday’s law of electromagnetic induction and

(iv) Ampere-Maxwell law.

The last one is the well known Ampere’s law with Maxwell’s modification for incorporating displacement current (which can flow even through empty space), in addition to conduction current (which flow through conductors). Here are Maxwell’s equations:

1. ∫_{closed surface}E.dA = Q/ε₀

[This is Gauss’s law in electricity which states that the flux of the electric field E through any closed surface, that is, the surface integral of E.dA over any closed surface is 1/ε₀ times the total charge Q enclosed by the surface. Note that E is the electric field vector present at an elemental area vector dA of the closed surface].

2. ∫_{closed surface}B.dA = 0

[This is Gauss’s law in magnetism which states that the magnetic flux through a closed surface is zero].

3. ∫_{closed path}E.dℓ = – dФ_B/dt

[This is Faraday’s law of electromagnetic induction which states that the induced emf in a circuit is equal to the time rate of change of magnetic flux Ф_B. The negative sign is because of Lenz’s law which states that the induced emf opposes the change of the magnetic flux].

4. ∫_{closed path}B.dℓ = μ₀i_c + μ₀ε₀dФ_E/dt

[This is Ampere’s circuital law with Maxwell’s modification. The firest term, μ₀i_c on the right hand side contains the conduction current i_c. The second term, μ₀ε₀(dФ_E/dt) was added by Maxwell to incorporate the displacement current i_d = ε₀(dФ_E/dt). Note that the displacement current is produced because of the time rate of change of the electric field].

Maxwell’s equations given above are in the integral form. The differential form of Maxwell’s equations can be easily obtained by applying Gauss’s divergence theorem and Stokes theorem. Thus we have the following equations:

(i) ∫_{closed surface}E.dA = Q/ε₀ becomes ∫_v div Edv = (1/ε₀) ∫_v ρdv, on applying Gauss divergence theorem to the left hand side of the equation and by putting Q = ∫_v ρdv where ρ is the volume charge density. The volume integration is done over the volume ‘v’ enclosed by the closed surface. Therefore,

div E = ρ/ε₀

This can be written also as

div D = ρ

where D is the electric displacement vector which in free space is given by D = ε₀E.

(ii) The second equation, ∫_{closed surface}B.dA = 0 similarly becomes

div B = 0

(iii) The 3^rd equation, ∫_{closed path} E.dℓ = – dФ_B/dt becomes ∫_s curl E.dA = ∫_s (d/dt) B. dA, on applying Stokes theorem to the left hand side of the equation and by putting the magnetic flux Ф_B = ∫_s B. dA. The surface integration is performed over the entire area A enclosed by the closed path. Therefore,

curl E = – dB/dt

(iv) The 4^th equation, ∫_{closed path}B.dℓ = μ₀i_c + μ₀ε₀ (dФ_E/dt) becomes ∫_s curl B.dA = ∫_s μ₀J. dA + ∫_s μ₀ε₀(d/dt) E. dA, on applying Stokes theorem to the left hand side of the equation and by putting i_c = ∫_s J . dA and Ф_E= ∫_s E. dA. Therefore,

curl B = μ₀J + μ₀ε₀(dE/dt) = μ₀ (J + dD/dt) since D = ε₀E for free space.

Displacement current:

The concept of the displacement current was introduced by Maxwell from his understanding that all electric currents must be closed. For instance, in the charging of a capacitor, a conduction current i_c flows in the wires connecting the capacitor to the charging battery and an equal (total) displacement current flows through the dielectric (or free space) in between the capacitor plates. The conduction current in the connecting wire and the displacement current in the space between the plates of the capacitor make a closed current circuit.

The displacement current (i) as well as the conduction current is given, as usual, by

i = dQ/dt

But Q = CV where C is the capacitance and V is the voltage across the capacitor. Therefore dQ = CdV so that the displacement current i = CdV/dt.

In the simple case of a parallel plate capacitor with air or free space between the plates, C = ε₀A/d where A is the area of the plates and d is the separation between the plates. Further, V = Ed where E is the electric field between the plates. Therefore, displacement current i = (ε₀A/d)×d×dE/dt = ε₀A dE/dt = ε₀dФ_E/dt, on substituting for the electric flux Ф_E = AE.

The above steps show that the quantity ε₀dФ_E/dt indeed represents the displacement current.

Electromagnetic Waves:

Electromagnetic waves are produced by accelerated charges. An oscillating electric charge produces an oscillating electric field in space, which produces an oscillating magnetic field. But an oscillating magnetic field is a source of oscillating electric field. Therefore an oscillating electric charge can produce oscillating electric and magnetic fields which regenerate each other and an electromagnetic wave propagates through the space.

The electric field and the magnetic field in an electromagnetic wave are perpendicular to each other. (Note that in charging a capacitor, the electric field in the space between the capacitor plates is directed perpendicular to the plates where as the magnetic field produced by the displacement current flowing through the space between the capacitor plates is along circles around the electric field lines).

In the case of a plane electromagnetic wave propagating along the positive z-direction, the electric field is along the positive x-direction and the magnetic field is along the positive y-direction.

Remember:

(i) The vectors E and B in an electromagnetic wave are perpendicular to each other and so oriented that the vector product (cross product) E×B points in the direction of propagation of the wave.

As an example of the application of this rule, suppose the electric field vector is along the negative z-direction and the magnetic field vector is along the positive x-direction. This wave has to be propagating along the negative y-direction (Fig.).

(ii) The magnitudes (E and B) of the electric field and the magnetic field in an electromagnetic wave are related as

B = E/c

where c is the speed of electromagnetic waves.

(iii) The speed (c) of electromagnetic waves in free space is given by

c = 1/√(μ₀ ε₀)

(iii) The speed (v) of electromagnetic waves in a medium of permittivity ε and permeability μ is given by

v = 1/√(μ ε)

Since μ = μ₀ μ_r and ε = ε₀ ε_r where μ_r and ε_r are respectively the relative permeability and the relative permittivity of the medium, we have

v = c/√(μ_r ε_r)

(iv) The energy density (energy per unit volume) in the region of space through which an electromagnetic wave propagates is due to the electric and magnetic fields associated with the wave. The energy density (U_E) due to the electric field is given by

U_E = ½ ε₀E_rms² where E_rms is the root mean square value of the electric field. [We use the root mean square value since the field is oscillating].

Similarly the energy density (U_B) due to the magnetic field is given by

U_B = ½ (B_rms²/μ₀) where B_rms is the root mean square value of the magnetic field.

U_E and U_B are equal since B_rms = E_rms/c = E_rms√(μ₀ ε₀)

The energy density (U) due to the electromagnetic wave is the sum of the above energy densities and is given by

U = U_E + U_B = ½ ε₀E_rms² + ½ (B_rms²/μ₀)

Since U_E = U_B, we have

U = ε₀E_rms² = B_rms²/μ₀

In the case sinusoidally oscillating electric and magnetic fields E_rms = E_m/√2 and B_rms = B_m/√2 so that the energy density can be written as

U = ε₀E_m²/2 = B_m²/2μ₀

(v) The intensity (I) of the electromagnetic wave, which is the power flow through unit area, that is, the energy flowing per second through unit area (the plane of the area held perpendicular to the direction of propagation of the wave) is given by

I = Uc = ε₀c E_m²/2 = (E_m²/2)√(ε₀/μ₀) since c =1/√(μ₀ ε₀)

This can also be written as

I = E_m²/2cμ₀

(vi) The power flow through unit area is described by Poynting vector S given by

S = E×B/μ₀

Poynting vector S is directed along the direction of propagation of the wave. Since E and B are perpendicular to each other, Poynting vector has magnitude EB/μ₀ which is equal to E²/cμ₀. Note that this quantity represents the instantaneous power flow through unit area.

In the next post we will discuss questions in this section. Meanwhile see these posts at physicsplus.