JEE Physics Electromagnetic Waves Notes Set 02

Displacement Current

Between the plates of a charging capacitor no charge crosses the gap, yet the magnetic field must still be produced. Maxwell resolved this by adding a displacement current, defined by \( I_d = \varepsilon_0\frac{d\Phi_E}{dt} \). Total current is \( I = I_c + I_d \), and Ampere's law becomes \( \oint\vec{B}\cdot d\vec{l} = \mu_0(I_c + I_d) \). For a capacitor with dielectric constant K, the displacement current equals the charging current \( C\frac{dV}{dt} \), with \( C = \frac{K\varepsilon_0A}{d} \).

Nature of Electromagnetic Waves

1. A changing electric field produces a changing magnetic field and vice versa, so the two sustain each other and travel through space without any medium.
2. The waves are transverse. \( \vec{E} \), \( \vec{B} \) and the direction of travel are mutually perpendicular, and the direction of propagation is that of \( \vec{E}\times\vec{B} \). For a wave along +z, \( \vec{E} \) along x goes with \( \vec{B} \) along y.
3. \( \vec{E} \) and \( \vec{B} \) are in phase and \( \frac{E_0}{B_0} = c \).
4. Speed in vacuum \( c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3\times10^8 \) m/s. In a medium \( v = \frac{1}{\sqrt{\mu\varepsilon}} = \frac{c}{n} \), with \( n = \sqrt{\varepsilon_r\mu_r} \).
5. On entering a medium the frequency is unchanged, while speed and wavelength reduce by the factor n.

Energy and Momentum

1. Energy density in the electric field \( \frac{1}{2}\varepsilon_0E^2 \) equals that in the magnetic field \( \frac{B^2}{2\mu_0} \). Average energy density of each is \( \frac{1}{4}\varepsilon_0E_0^2 \), and total average is \( \frac{1}{2}\varepsilon_0E_0^2 \).
2. Intensity \( I = \frac{1}{2}\varepsilon_0E_0^2c \).
3. Radiation carries momentum, so it exerts pressure. Energy U absorbed brings momentum \( \frac{U}{c} \). For intensity I on area A, force \( \frac{IA}{c} \) if fully absorbed and \( \frac{2IA}{c} \) if fully reflected. If a fraction \( \eta \) is absorbed and the rest reflected, force is \( \frac{IA}{c}(2 - \eta) \).

Electromagnetic Spectrum

All have the same speed in vacuum. Alpha and beta rays are streams of particles, not electromagnetic waves.

Worked Examples

1. Electric amplitude 480 V/m gives \( B_0 = \frac{480}{3\times10^8} = 1.6\times10^{-6} \) T.
2. A wave of 3 MHz entering a medium with \( \varepsilon_r = 4 \) (and \( \mu_r = 1 \)) keeps its frequency while \( n = 2 \), so the wavelength is halved.
3. A parallel plate capacitor of area 1 m², gap 1 mm and K = 10 is charged at 25 V/s: \( I_d = C\frac{dV}{dt} = 8.85\times10^{-8}\times25 \approx 2.2\ \mu A \).