Displacement Current and the Nature of Electromagnetic Waves: Quick Revision

A brief note on Displacement Current and the Nature of Electromagnetic Waves. Read it before a test. Try each question at the end before you open its answer.

Displacement current

• The puzzle: a flat disc and a bowl with the same rim give different currents in Ampere's law for a charging capacitor.
• Displacement current: $I_d = \varepsilon_0\,d\Phi_E/dt$. For a charging capacitor, $I_d = I_c$. For parallel plates, $I_d = C\,dV/dt$.
• Ampere-Maxwell law: $\oint \vec{B}\cdot d\vec{l} = \mu_0(I_c + I_d)$.
• Circular plates of radius $R$: $B = \mu_0 I r/2\pi R^2$ for $r < R$, and $B = \mu_0 I/2\pi r$ for $r > R$.
• It is a changing electric flux, not a flow of charge.

Maxwell's equations

• Gauss, electric: charge makes $\vec{E}$. $\oint \vec{E}\cdot d\vec{A} = Q/\varepsilon_0$.
• Gauss, magnetic: no isolated poles. $\oint \vec{B}\cdot d\vec{A} = 0$.
• Faraday: changing $B$ makes $E$. $\oint \vec{E}\cdot d\vec{l} = -d\Phi_B/dt$.
• Ampere-Maxwell: current and changing $E$ make $B$.
• Speed: $c = 1/\sqrt{\mu_0\varepsilon_0} = 3.00 \times 10^{8}$ m/s. In a material, $v = c/\sqrt{\mu_r\varepsilon_r}$.

The plane wave

• Form: $E = E_0\sin(kz - \omega t)$ and $B = B_0\sin(kz - \omega t)$, with $k = 2\pi/\lambda$ and $\omega = 2\pi f$.
• Properties: transverse, $\vec{E}\perp\vec{B}$, in phase, travelling along $\vec{E}\times\vec{B}$, no medium needed.
• Strength: $E_0 = cB_0$ in vacuum. $c = f\lambda$.
• In a material: $f$ unchanged, $v = c/n$, $\lambda = \lambda_0/n$.
• Source: an accelerated charge. Hertz made the waves in 1887.

A worked example

A plane wave in vacuum has $E_0 = 54$ V/m and a frequency of $80$ MHz. Find the wavelength and the peak magnetic field.

1. Wavelength. $\lambda = \dfrac{c}{f} = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{80 \times 10^{6}\ \text{Hz}} = 3.750$ m.
2. Magnetic field. $B_0 = \dfrac{E_0}{c} = \dfrac{54\ \text{V/m}}{3.00 \times 10^{8}\ \text{m/s}} = 180$ nT.

Practise with new numbers

Check yourself

1. Why must Maxwell's term be added to Ampere's law?

2. In a plane wave travelling along $+z$, the electric field is along $+x$. Where does the magnetic field point?

3. What is needed for a charge to make an electromagnetic wave?

Sources

• OpenStax, University Physics Volume 2, Chapter 16 (Electromagnetic Waves).
• NCERT Physics Class 12, Chapter 8 (Electromagnetic Waves).