The electric field in an electromagnetic wave is given by E = (50 NC⁻¹) sinω (t - x/c)
The energy contained in a cylinder of volume V is 5.5 × 10⁻¹² J. The value of V is _____________ cm³. (given ∈₀ = 8.8 × 10⁻¹²C²N⁻¹m⁻²)

Solution

Physics sir · 3 steps

Find the average energy density from the peak field, then volume is just energy divided by that density.

1. Average energy density. In an EM wave the electric and magnetic fields carry equal energy. Averaged over time, each contributes 1/4ε₀ E₀², so the total is 1/2ε₀ E₀², where E₀ = 50 N C⁻¹ is the peak field.

   uₐᵥ = 1/2ε₀ E₀² = 1/2 × 8.8 × 10⁻¹² × (50)² = 1.1 × 10⁻⁸ J m⁻³

2. Find the volume. V = U/uₐᵥ = (5.5 × 10⁻¹²)/(1.1 × 10⁻⁸) = 5 × 10⁻⁴ m³.
3. So. Since 1 m³ = 10⁶ cm³, V = 5 × 10⁻⁴ × 10⁶ = 500 cm³.

The common slip: 1000. Counting only the electric field energy, 1/4ε₀ E₀², halves the energy density and gives 1000 instead of 500.

The idea to keep. Total average energy density of an EM wave is 1/2ε₀ E₀²: half comes from the electric field and half from the magnetic field. Never forget the magnetic half.

Answer from the NTA final answer key. Solution by PrepWiser.