CBSE · Class 12 · Physics · Chapter 8

Electromagnetic Waves — Formula Sheet

Board Formulas

10 formulas

1.Displacement Current★
$i_{d} = ε_{0} \frac{d Φ _{E}}{d t}$
$i_{d}$ : Displacement current (A) · $ε_{0}$ : Permittivity of free space = 8.854×10⁻¹² C² N⁻¹ m⁻² · $Φ_{E}$ : Electric flux (N m² C⁻¹)
A current associated with a changing electric flux, e.g. between the plates of a charging capacitor. No charge actually flows across the gap. Unit: ampere (A).
2.Displacement Current in a Parallel-Plate Capacitor
$i_{d} = ε_{0} A \frac{d E}{d t} = C \frac{d V}{d t}$
$A$ : Plate area (m²) · $C$ : Capacitance (F) · $\frac{d V}{d t}$ : Rate of change of voltage across the plates (V s⁻¹)
Uses Φ_E = EA, E = V/d and C = ε₀A/d. Inside a charging capacitor i_d equals the conduction current in the connecting wires.
3.Ampère–Maxwell Law★
$\oint B \cdot d l = μ_{0} i_{c} + μ_{0} ε_{0} \frac{d Φ _{E}}{d t}$
$i_{c}$ : Conduction current (A) · $μ_{0}$ : Permeability of free space = 4π×10⁻⁷ T m A⁻¹
The source of B is the total current: conduction current i_c plus displacement current i_d. A changing electric field produces a magnetic field.
4.Maxwell's Equations
$\oint E \cdot d A = \frac{Q}{ε _{0}} \oint B \cdot d A = 0 \oint E \cdot d l = - \frac{d Φ _{B}}{d t} \oint B \cdot d l = μ_{0} i_{c} + μ_{0} ε_{0} \frac{d Φ _{E}}{d t}$
In order: Gauss's law for electricity, Gauss's law for magnetism, Faraday's law and the Ampère–Maxwell law. Together they predict electromagnetic waves.
5.Speed of EM Waves in Vacuum★
$c = \frac{1}{μ _{0} ε _{0}} \approx 3 \times 1 0^{8} m s^{- 1}$
$c$ : Speed of light in vacuum (m s⁻¹)
Maxwell obtained this speed from purely electric and magnetic constants. It matched the measured speed of light, showing that light is an electromagnetic wave.
6.Speed of EM Waves in a Medium
$v = \frac{1}{μ ε}$
$v$ : Speed in the medium (m s⁻¹) · $μ$ : Permeability of the medium (T m A⁻¹) · $ε$ : Permittivity of the medium (C² N⁻¹ m⁻²)
μ and ε are the permeability and permittivity of the medium. With μ = μ_rμ₀ and ε = ε_rε₀, the refractive index is n = c/v = √(μ_rε_r).
7.Plane EM Wave Travelling Along z
$E_{x} = E_{0} sin (k z - ω t), B_{y} = B_{0} sin (k z - ω t)$
$E_{0}$ : Electric field amplitude (V m⁻¹) · $B_{0}$ : Magnetic field amplitude (T) · $k$ : Wave number (rad m⁻¹) · $ω$ : Angular frequency (rad s⁻¹)
E along x, B along y, propagation along +z. E and B oscillate in phase and are perpendicular to each other and to the direction of travel, so the wave is transverse.
8.Ratio of Field Amplitudes★
$\frac{E _{0}}{B _{0}} = c$
Also true at every instant: E = cB. B is numerically tiny: E₀ = 3 V m⁻¹ corresponds to B₀ = 10⁻⁸ T.
9.Wave Number and Angular Frequency
$k = \frac{2 π}{λ}, ω = 2 π ν, ω = c k$
$λ$ : Wavelength (m) · $ν$ : Frequency (Hz)
Dividing ω by k gives the wave speed: ω/k = νλ = c.
10.Speed, Frequency and Wavelength
$c = ν λ$
All EM waves travel at c in vacuum. In a medium ν stays the same while both speed and wavelength drop by the factor n.