"Electromagnetic Waves" — Practice

Question 1

Why did Ampere's original law need modification, and what did Maxwell add?

Accepted Answer

For a charging capacitor, two surfaces sharing the same boundary loop gave different enclosed currents: a flat one pierced by the wire gave $I$, while a bag-shaped one passing between the plates gave zero. Maxwell added the displacement current $\varepsilon_0\,d\Phi_E/dt$, which equals $I$ in the gap, so both surfaces now agree.

Question 2

Arrange gamma rays, radio waves, visible light, X-rays, infrared, microwaves and ultraviolet in order of increasing wavelength.

Accepted Answer

Gamma rays, X-rays, ultraviolet, visible, infrared, microwaves, radio waves. "Infra" means below, so infrared is below red in frequency and above it in wavelength, while ultraviolet is above violet in frequency.

Question 3

In a travelling electromagnetic wave, what is the phase relationship between the electric and magnetic fields?

Accepted Answer

They are exactly in phase, reaching their maxima and their zeros at the same instant and the same place. They do not alternate, and they carry equal energy on average despite $B_0$ being numerically $3	imes10^{8}$ times smaller in SI units.

Question 4

A capacitor has circular plates of radius 6 cm and the field between them rises at $3	imes10^{10}$ V m$^{-1}$ s$^{-1}$. Find the displacement current.

Accepted Answer

The plate area is $\pi(0.06)^{2} \approx 1.13	imes10^{-2}$ m$^{2}$, and $I_d = \varepsilon_0A\,dE/dt \approx 3.0	imes10^{-3}$ A, that is 3 mA. This is exactly the conduction current in the feeding wires.

Question 5

A plane electromagnetic wave has electric field amplitude 120 N C$^{-1}$. Find the magnetic field amplitude and the average energy density.

Accepted Answer

$B_0 = E_0/c = 4	imes10^{-7}$ T. The average energy density is $	frac{1}{2}\varepsilon_0E_0^{2} \approx 6.37	imes10^{-8}$ J m$^{-3}$, counting both fields since they contribute equally.

Question 6

Light of wavelength 600 nm in vacuum enters glass of refractive index 1.5. Find the speed, wavelength and frequency inside the glass.

Accepted Answer

The speed becomes $2	imes10^{8}$ m s$^{-1}$ and the wavelength 400 nm, both reduced by the factor 1.5. The frequency is unchanged at $5	imes10^{14}$ Hz, because it is fixed by the source — which is also why the colour does not change.

Question 7

Sunlight arrives at about 1000 W m$^{-2}$. Find the peak electric field and the radiation pressure on a perfectly absorbing surface.

Accepted Answer

From $I = 	frac{1}{2}\varepsilon_0E_0^{2}c$, the peak field is about 868 N C$^{-1}$. The radiation pressure is $I/c = 3.3	imes10^{-6}$ Pa, roughly $3	imes10^{-11}$ atmospheres. On a perfect reflector it would double to $6.7	imes10^{-6}$ Pa.

Question 8

Explain why the value of $c$ computed from $\mu_0$ and $\varepsilon_0$ was such a significant result.

Accepted Answer

Both constants come from purely electrical and magnetic measurements — $\varepsilon_0$ from forces between static charges, $\mu_0$ from forces between currents. Neither involves light in any way. When $1/\sqrt{\mu_0\varepsilon_0}$ turned out to equal the independently measured speed of light, the only reasonable conclusion was that light is itself an electromagnetic wave. The identification was made by arithmetic, more than twenty years before Hertz produced such waves deliberately.

Question 9

Why does the biological hazard of electromagnetic radiation rise with frequency rather than with intensity alone?

Accepted Answer

Because the damage mechanism changes. Photon energy is proportional to frequency, so low-frequency radio and microwave photons can only deposit energy as heat, however many arrive. Above a threshold inside the ultraviolet band, a single photon carries enough to break a chemical bond and damage DNA directly, which no amount of lower-frequency radiation can do. This is why UV is subdivided into A, B and C with sharply different health implications.

Question 10

Why can light be polarised but sound cannot, and what does this prove?

Accepted Answer

Polarisation means selecting one plane of vibration out of the many available perpendicular to the direction of travel. A transverse wave has such planes to choose between; a longitudinal wave vibrates only along the direction of travel and has none. Since light can be polarised and sound cannot, light must be transverse and sound longitudinal — historically one of the decisive pieces of evidence for the transverse nature of light.

Practice — "Electromagnetic Waves"

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