Electromagnetic Waves

Notes

1. Electromagnetic Waves

The name electromagnetic wave comes from electric and magnetic waves.

• A time-varying electric field produces a magnetic field. This is given by Faraday.
• A time-varying magnetic field produces an electric field. This is given by Maxwell.

Hence, the combination or couple of time-varying electric field and magnetic field produces electromagnetic waves.

A time-varying current produces a magnetic field.

A time-varying electric field produces displacement current.

A time-varying magnetic field produces current.

The total current is:

I = I_c + I_d

where:

• I_c = conduction current
• I_d = displacement current

The displacement current is:

I_d = ε₀ dΦ_E/dt

For alternating current:

I = I₀ sinωt

Electromagnetic waves are waves in which the electric field E and magnetic field B are perpendicular to each other. Both are also perpendicular to the direction of propagation of the wave.

2. Inconsistency of Ampere's Circuital Law or Maxwell's Assumption

According to Ampere's Circuital Law:

∮ B · dl = μ₀I

Consider a capacitor and assume two Amperian loops.

• C₁: Just to the left side of the capacitor plate.
• C₂: Just to the right of the capacitor plate.

For Loop C₁

Applying Ampere's Circuital Law:

∮ B · dl = μ₀I

For Loop C₂

No conduction current passes through the space between the capacitor plates.

Therefore:

∮ B · dl = μ₀ × 0

∮ B · dl = 0

Thus, the two results are inconsistent.

This shows the inconsistency of Ampere's Circuital Law for a capacitor.

3. Ampere-Maxwell Law – Improvement of Ampere's Circuital Law

To correct Ampere's Circuital Law, Maxwell considered a capacitor of capacitance C.

Let:

• d = separation between the plates
• I = flowing current
• V = potential difference across the plates

Since positive and negative charges exist on the two plates respectively, an electric field exists between the plates.

The electric field between the plates is:

E = V/d

We know:

C = Q/V

Therefore:

V = Q/C

Hence:

E = (Q/C)/d

For a parallel plate capacitor:

C = Aε₀/d

Therefore:

E = Q/(Aε₀)

Hence:

Q = EAε₀

Since the capacitor gets charged and discharged, the amount of charge stored by the capacitor is variable.

Differentiating with respect to time:

dQ/dt = ε₀ d(EA)/dt