10 worked answers · short steps first, explanations when you need them
Chapter 8 Exercises
Textbook pages 213–214
Question 1
A capacitor has circular plates of radius 12 cm separated by 5.0 cm and is charged at 0.15 A. Find (a) capacitance and rate of change of voltage, (b) displacement current, and (c) whether Kirchhoff's junction rule applies at a plate.
Steps only
C=ε₀πr²/d≈8.0 pF; dV/dt=I/C≈1.9×10¹⁰ V/s; Id=I=0.15 A
Answer
8.0 pF; 1.9×10¹⁰ V/s; 0.15 A
Question 2
A circular-plate capacitor has plate radius 6.0 cm and capacitance 100 pF. It is connected to a 230 V rms, 300 rad/s AC supply. Find (a) rms conduction current, (b) whether it equals displacement current, and (c) magnetic-field amplitude 3.0 cm from the axis between the plates.
Steps only
Irms=ωCV=6.9 μA; I₀=9.76 μA; B₀=μ₀I₀r/(2πR²)=1.6×10⁻¹¹ T
Answer
6.9 μA; equal currents; 1.6×10⁻¹¹ T
Question 3
What physical quantity is the same for X-rays of wavelength 10⁻¹⁰ m, red light of wavelength 6800 Å and radio waves of wavelength 500 m?
Steps only
All electromagnetic waves travel at c=3.0×10⁸ m/s in vacuum.
Answer
Speed: 3.0×10⁸ m/s
Question 4
A plane electromagnetic wave travels in vacuum along the z-direction. State the directions of its electric and magnetic fields. If its frequency is 30 MHz, find its wavelength.
Steps only
E⊥B⊥z; λ=c/f=(3×10⁸)/(30×10⁶)=10 m
Answer
Mutually perpendicular fields; 10 m
Question 5
A radio tunes to frequencies from 7.5 MHz to 12 MHz. Find the corresponding wavelength band.
Steps only
λ=c/f: 12 MHz→25 m; 7.5 MHz→40 m
Answer
25–40 m
Question 6
A charged particle oscillates at 10⁹ Hz. What is the frequency of the electromagnetic waves it produces?
Steps only
Wave frequency equals the oscillator frequency.
Answer
10⁹ Hz
Question 7
An electromagnetic wave in vacuum has magnetic-field amplitude 510 nT. Find its electric-field amplitude.
Steps only
E₀=cB₀=(3×10⁸)(510×10⁻⁹)=153 V/m
Answer
153 V/m
Question 8
An electromagnetic wave has electric-field amplitude 120 N/C and frequency 50.0 MHz. Find (a) B₀, angular frequency, wave number and wavelength, and (b) expressions for E and B.
Steps only
B₀=E₀/c=4×10⁻⁷ T; ω=2πf=3.14×10⁸; λ=6 m; k=2π/λ=1.05 rad/m
Answer
For +z travel: E=120 sin(kz−ωt)x̂; B=4×10⁻⁷ sin(kz−ωt)ŷ
Question 9
Use E=hf to estimate photon energies in eV for the different electromagnetic-spectrum regions. How are their energy scales related to their sources?
Steps only
E(eV)=hc/λ≈1.24×10⁻⁶/λ(m); shorter λ means higher photon energy.
Answer
Energy order: radio < microwave < IR < visible < UV < X-ray < gamma
Question 10
A plane wave has frequency 2.0×10¹⁰ Hz and electric-field amplitude 48 V/m. Find (a) wavelength, (b) magnetic-field amplitude and (c) show that average electric and magnetic energy densities are equal.
Steps only
λ=c/f=0.015 m; B₀=E₀/c=1.6×10⁻⁷ T; ⟨uE⟩=ε₀E₀²/4=⟨uB⟩≈5.1×10⁻⁹ J/m³