8. Electromagnetic Waves

Big idea: one linked electric–magnetic disturbance can carry energy across empty space.

Connect the charging-capacitor argument to the orientation, speed and spectrum of waves.

James Clerk Maxwell combined established electric and magnetic laws with a changing-electric-field contribution in the nineteenth century. Heinrich Hertz later produced and detected radio waves, supporting the electromagnetic-wave description.

Light, radio and X-rays belong to one family even though their sources and interactions look different.

An electromagnetic wave carries coupled electric and magnetic fields through space. In vacuum it needs no material medium. This differs from a sound wave, which requires matter whose pressure and density can oscillate.

The central link is change: a changing magnetic field is associated with a circulating electric field, and a changing electric field contributes to a circulating magnetic field. Maxwell’s equations combine these statements with the laws governing charges and magnetic flux. Their wave solutions identify light as an electromagnetic phenomenon.

A spacecraft sends a radio message across an almost empty region. Does lack of air prevent the signal?

Radio is an electromagnetic wave, so a material medium is not required for propagation.

Result: No. The wave can propagate through vacuum. Sound could not carry the same message through empty space.

1. Which field pair occurs in an electromagnetic wave?
2. What distinguishes vacuum light from ordinary sound?

Answers

1. Electric and magnetic fields.
2. Light can propagate without matter; sound requires a material medium.

A charging capacitor exposes a missing term in the steady-current form of Ampère’s law.

Draw one loop around a capacitor lead. A spanning surface cutting the wire encloses conduction current. Another surface with the same boundary can bulge between the capacitor plates, where no conduction charge crosses an ideal insulating gap. The magnetic circulation cannot depend on the imaginary surface you choose.

Maxwell supplied a displacement-current contribution $I_d=\epsilon_0\,d\Phi_E/dt$ in vacuum. The corrected law is $\oint\mathbf B\cdot d\mathbf l=\mu_0(I_c+\epsilon_0\,d\Phi_E/dt)$. The electric flux is $\Phi_E=\int\mathbf E\cdot d\mathbf A$. A changing field is the essential condition; a static electric field contributes no displacement current.

For an ideal parallel-plate capacitor, neglect fringing: $E=Q/(\epsilon_0A)$ and $\Phi_E=EA=Q/\epsilon_0$. Therefore $I_d=\epsilon_0\,d(Q/\epsilon_0)/dt=dQ/dt=I_c$. Across the gap use the displacement term; through the lead use conduction current. Do not count the full lead current twice for one selected surface.

A uniform electric field between vacuum plates changes at $5.0\times10^8\,\mathrm{V\,m^{-1}\,s^{-1}}$ over $0.020\,\mathrm{m^2}$. Use $\epsilon_0=8.854\times10^{-12}\,\mathrm{F\,m^{-1}}$. Find displacement current.

$I_d=\epsilon_0A\,dE/dt$.

Result: $I_d=0.000089\,\mathrm A$.

Displacement current has amperes as its unit but is not necessarily transport of free charges across the gap. It restores a surface-independent magnetic law consistent with charge conservation. In material media a fuller description includes polarization; the vacuum expression here is sufficient for the ideal capacitor model.

For uniform changing field over circular plates of radius R, a concentric loop of radius r inside the plate area encloses displacement current $I_d r^2/R^2$. Symmetry then gives $B(2\pi r)=\mu_0I_d r^2/R^2$, or $B=\mu_0I_dr/(2\pi R^2)$ for r<R. Outside the plate area, ignoring fringing and return-current details, the loop encloses the full displacement current and $B=\mu_0I_d/(2\pi r)$. The interior field grows with r instead of diverging at the axis.

1. When is displacement current zero?
2. What equals the conduction current while an ideal capacitor charges?

Answers

1. When electric flux through the selected surface is constant.
2. The displacement current through a suitable surface spanning the gap, neglecting fringing.

A plane wave makes the geometry of electromagnetic propagation especially clear.

Accelerated charges can emit electromagnetic radiation. A charge oscillating in an antenna is one familiar source. Uniform motion in an inertial frame is not the same as acceleration; a static charge has an electric field but does not emit a continuing radiation wave in that frame.

Maxwell’s vacuum equations predict $c=1/\sqrt{\mu_0\epsilon_0}$. For school calculations use $c\approx3.00\times10^8\,\mathrm{m\,s^{-1}}$. Frequency f, wavelength λ and speed obey $c=f\lambda$ in vacuum. In matter the phase speed may differ; frequency stays fixed when a stationary boundary is crossed, while wavelength changes.

For a vacuum plane wave travelling along +x, one possible choice is $\mathbf E=E_0\sin(kx-\omega t)\,\hat{\mathbf y}$ and $\mathbf B=B_0\sin(kx-\omega t)\,\hat{\mathbf z}$. Both fields are perpendicular to propagation and to each other. They are in phase, and $E_0=cB_0$. Their cross product specifies the energy-flow direction.

A vacuum radio wave has frequency $75\,\mathrm{MHz}$. Find its wavelength.

Convert MHz to hertz, then use $\lambda=c/f$.

Result: $\lambda=4\,\mathrm m$.

The electric-field amplitude is $90\,\mathrm{V\,m^{-1}}$. Find magnetic-field amplitude.

$B_0=E_0/c$.

Result: $B_0=3e-7\,\mathrm T$.

Instantaneous vacuum energy densities are $u_E=\epsilon_0E^2/2$ and $u_B=B^2/(2\mu_0)$. The relation E=cB makes these equal in a plane wave. For a sinusoidal wave, the average total density is $\langle u\rangle=\epsilon_0E_0^2/2$. Average intensity is $c\langle u\rangle$.

For the field amplitude above, $\langle u\rangle=3.5859e-8\,\mathrm{J\,m^{-3}}$. A wave also transports momentum. Normal absorption gives radiation pressure $I/c$; ideal normal reflection gives $2I/c$. These expressions assume an ideal plane beam and the stated boundary behaviour.

1. If E points +y and B points +z, where does energy flow?
2. What is the phase difference between E and B in a vacuum travelling plane wave?
3. Can a static charge be called a radiation source merely because it has a field?

Answers

1. Along +x because y cross z is x.
2. Zero.
3. No. A static field and outgoing radiation are different situations.

The spectrum is a continuous range of frequencies; category names summarize common sources and interactions.

From long wavelength to short wavelength, the usual order is radio, microwave, infrared, visible, ultraviolet, X-ray and gamma. Frequency and photon energy increase along that order while vacuum wavelength decreases. The boundaries are conventional and sometimes overlap; X-ray and gamma names can also reflect how photons are produced.

Visible light occupies only a small part of the spectrum. Within it, red has longer wavelength than violet in vacuum. A hot object emits a broad spectrum rather than only one infrared wavelength. Infrared is not itself “heat”: absorption can transfer wave energy into thermal energy.

The atmosphere transmits some bands better than others. This affects communication and astronomical observation. A short wavelength does not automatically mean a wave passes through every material; absorption depends on the material and interaction mechanisms.

Compare a $0.060\,\mathrm m$ microwave with a $600\,\mathrm{nm}$ visible wave. Use the same vacuum speed.

Frequency is inversely proportional to wavelength. The wavelength ratio is $0.060/(600\times10^{-9})$.

Result: The visible frequency is $100000 $ times the microwave frequency.

All these waves obey the same vacuum propagation speed. Their different effects follow from frequency, photon energy, intensity and interactions with matter. Neither speed alone nor the brightness seen by an eye can rank the risks of invisible radiation.

1. Which has greater vacuum wavelength: infrared or ultraviolet?
2. Why is there no single sharp physical boundary separating every spectrum category?
3. What remains equal for radio and gamma waves in vacuum?

Answers

1. Infrared.
2. The spectrum is continuous and several labels are conventional or source-based.
3. Their propagation speed c.

• Displacement current in vacuum: $I_d=\epsilon_0d\Phi_E/dt$.
• Plane vacuum waves: $c=1/\sqrt{\mu_0\epsilon_0}$, $c=f\lambda$, $E_0=cB_0$.
• E, B and propagation are mutually perpendicular; E and B oscillate in phase.
• Frequency rises from radio through microwave, infrared, visible, ultraviolet, X-ray to gamma.

• A field can exist without a propagating radiation wave.
• Spectrum labels are useful categories, not separate kinds of vacuum propagation.
• The displacement-current term preserves consistency; it is not conduction through an insulating gap.

Try these newly written questions before reading the answers. Show a diagram, the governing equation, substitutions and units where appropriate.

8.1 Explain the capacitor-surface problem addressed by Maxwell’s correction.

8.2 Vacuum electric flux grows at $2.0\times10^9\,\mathrm{V\,m\,s^{-1}}$. Find displacement current using the stated epsilon value.

8.3 During steady ideal capacitor charging at 0.40 A, what is the displacement current in its gap?

8.4 Find wavelength at $125\,\mathrm{MHz}$.

8.5 Find vacuum frequency for $0.015\,\mathrm m$ wavelength.

8.6 Find B amplitude if E amplitude is $150\,\mathrm{V\,m^{-1}}$.

8.7 A wave travels +z and E points +x. Which direction does B point?

8.8 Does a plane wave carry more electric-field energy than magnetic-field energy in vacuum?

8.9 Arrange infrared, gamma and radio by increasing frequency.

8.10 What changes when monochromatic light enters a stationary transparent medium?

8.11 Why is an oscillating charge relevant to radiation?

8.12 A wave is completely reflected at normal incidence. Compare pressure with absorption at the same intensity.

Answers

8.1 Different spanning surfaces of the same loop must predict the same magnetic circulation; changing electric flux accounts for the gap surface.

8.2 $0.017708\,\mathrm A$.

8.3 0.40 A in the ideal parallel-plate approximation.

8.4 $2.4\,\mathrm m$.

8.5 $20000000000\,\mathrm{Hz}$.

8.6 $5e-7\,\mathrm T$.

8.7 +y, since x cross y points +z.

8.8 No. Their instantaneous energy densities are equal for the travelling plane-wave model.

8.9 Radio, infrared, gamma.

8.10 Its speed and wavelength can change; its frequency remains fixed.

8.11 Oscillation involves acceleration, which can produce outgoing electromagnetic radiation.

8.12 Ideal reflection gives twice the pressure because the normal momentum reverses.