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Ch. 2 Problems

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Basic

Medium

Basic

B-1. Numerical Calculation of the Speed of Light

Using \(\mu_0 = 4\pi \times 10^{-7} \;\text{T·m/A}\) and \(\varepsilon_0 = 8.854 \times 10^{-12} \;\text{F/m}\), calculate \(c = 1/\sqrt{\mu_0 \varepsilon_0}\) and compare the result with the measured value of the speed of light \(c = 2.998 \times 10^8 \;\text{m/s}\).

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B-2. From Potentials to Maxwell's Equations

Using the vector identity \(\nabla \cdot (\nabla \times \mathbf{A}) = 0\), show that \(\nabla \cdot \mathbf{B} = 0\) (Maxwell's second equation) follows automatically from the definition \(\mathbf{B} = \nabla \times \mathbf{A}\).

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Medium

M-1. Derivation of Electromagnetic Wave Speed

Starting from the 3rd and 4th of Maxwell's equations (in vacuum, \(\rho = 0\), \(\mathbf{j} = 0\)), derive the wave equation for the electric field \(\mathbf{E}\) and show that the wave speed is \(c = 1/\sqrt{\mu_0 \varepsilon_0}\).

Hint

Apply \(\nabla \times\) to both sides of the 3rd equation and use the vector identity \(\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}\).

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