Problems

Express the displacement current in a capacitor in terms of the capacitance and the rate of change of the voltage across the capacitor.

Suppose the parallel-plate capacitor shown below is accumulating charge at a rate of 0.010 C/s. What is the induced magnetic field at a distance of 10 cm from the capacitator?

A parallel-plate capacitor has a plate area of $A=0.250{\text{m}}^2$ and a separation of 0.0100 m. What must be must be the angular frequency $\omega$ for a voltage $V\left(t\right)=V_0\text{sin}\omega t$ with $V_0=100\text{V}$ to produce a maximum displacement induced current of 1.00 A between the plates?

The voltage across a parallel-plate capacitor with area A and separation d varies with time t as $V=a{t}^2$, where a is a constant. Find the displacement current between the plates.

What is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of $5.00\times {10}^{\mathrm{-4}}\text{T}$ (about 10 times Earth’s magnetic field)?

If electric and magnetic field strengths vary sinusoidally in time at frequency 1.00 GHz, being zero at $t=0$, then $E=E_0\text{sin}2\pi ft$ and $B=B_0\text{sin}2\pi ft$. (a) When are the field strengths next equal to zero? (b) When do they reach their most negative value? (c) How much time is needed for them to complete one cycle?

A plane electromagnetic wave of frequency 20 GHz moves in the positive y-axis direction such that its electric field is pointed along the z-axis. The amplitude of the electric field is 10 V/m. The start of time is chosen so that at $t=0$, the electric field has a value 10 V/m at the origin. (a) Write the wave function that will describe the electric field wave. (b) Find the wave function that will describe the associated magnetic field wave.

While outdoors on a sunny day, a student holds a large convex lens of radius 4.0 cm above a sheet of paper to produce a bright spot on the paper that is 1.0 cm in radius, rather than a sharp focus. By what factor is the electric field in the bright spot of light related to the electric field of sunlight leaving the side of the lens facing the paper?

The electric field of an electromagnetic wave is given by E =
$\left(6.0\times {10}^{\mathrm{-3}}\text{V/m}\right)\text{sin}\left[2\pi \left(\frac{x}{18\text{m}}-\frac{t}{6.0\times {10}^{\mathrm{-8}}\text{s}}\right)\right]\widehat{j}.$
Write the equations for the associated magnetic field and Poynting vector.

The filament in a clear incandescent light bulb radiates visible light at a power of 5.00 W. Model the glass part of the bulb as a sphere of radius $r_0=3.00\text{cm}$ and calculate the amount of electromagnetic energy from visible light inside the bulb.

An incandescent light bulb emits only 2.6 W of its power as visible light. What is the rms electric field of the emitted light at a distance of 3.0 m from the bulb?

A small helium-neon laser has a power output of 2.5 mW. What is the electromagnetic energy in a 1.0-m length of the beam?

The magnetic field of a plane electromagnetic wave moving along the z axis is given by $\overrightarrow{B}=B_0\text{cos}\left(kz+\omega t\right)\widehat{j}$, where $B_0=5.00\times {10}^{\mathrm{-10}}\text{T}$ and $k=3.14\times {10}^{\mathrm{-2}}{\text{m}}^{\mathrm{-1}}.$
(a) Write an expression for the electric field associated with the wave. (b) What are the frequency and the wavelength of the wave? (c) What is its average Poynting vector?

Assume the helium-neon lasers commonly used in student physics laboratories have power outputs of 0.500 mW. (a) If such a laser beam is projected onto a circular spot 1.00 mm in diameter, what is its intensity? (b) Find the peak magnetic field strength. (c) Find the peak electric field strength.

Suppose the maximum safe intensity of microwaves for human exposure is taken to be $1.00{\text{W/m}}^2$. (a) If a radar unit leaks 10.0 W of microwaves (other than those sent by its antenna) uniformly in all directions, how far away must you be to be exposed to an intensity considered to be safe? Assume that the power spreads uniformly over the area of a sphere with no complications from absorption or reflection. (b) What is the maximum electric field strength at the safe intensity? (Note that early radar units leaked more than modern ones do. This caused identifiable health problems, such as cataracts, for people who worked near them.)

Lasers can be constructed that produce an extremely high intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to initiate nuclear fusion, for example. Such a laser may produce an electromagnetic wave with a maximum electric field strength of $1.00\times {10}^{11}\text{V}\text{/}\text{m}$ for a time of 1.00 ns. (a) What is the maximum magnetic field strength in the wave? (b) What is the intensity of the beam? (c) What energy does it deliver on an $1.00{\text{-mm}}^2$ area?

What pressure does light emitted uniformly in all directions from a 100-W incandescent light bulb exert on a mirror at a distance of 3.0 m, if 2.6 W of the power is emitted as visible light?

A Styrofoam spherical ball of radius 2 mm and mass $20\text{μg}$ is to be suspended by the radiation pressure in a vacuum tube in a lab. How much intensity will be required if the light is completely absorbed by the ball?

Sunlight reaches the ground with an intensity of about $1.0{\text{kW/m}}^2$. A sunbather has a body surface area of $0.8{\text{m}}^2$ facing the sun while reclining on a beach chair on a clear day. (a) how much energy from direct sunlight reaches the sunbather’s skin per second? (b) What pressure does the sunlight exert if it is absorbed?

How many helium atoms, each with a radius of about 31 pm, must be placed end to end to have a length equal to one wavelength of 470 nm blue light?

Find the frequency range of visible light, given that it encompasses wavelengths from 380 to 760 nm.

Radio station WWVB, operated by the National Institute of Standards and Technology (NIST) from Fort Collins, Colorado, at a low frequency of 60 kHz, broadcasts a time synchronization signal whose range covers the entire continental US. The timing of the synchronization signal is controlled by a set of atomic clocks to an accuracy of $1\times {10}^{\mathrm{-12}}\text{s},$ and repeats every 1 minute. The signal is used for devices, such as radio-controlled watches, that automatically synchronize with it at preset local times. WWVB’s long wavelength signal tends to propagate close to the ground.
(a) Calculate the wavelength of the radio waves from WWVB.
(b) Estimate the error that the travel time of the signal causes in synchronizing a radio controlled watch in Norfolk, Virginia, which is 1570 mi (2527 km) from Fort Collins, Colorado.

7. The prefix “mega” (M) and “kilo” (k), when referring to amounts of computer data, refer to factors of 1024 or $2^{10}$ rather than 1000 for the prefix kilo, and ${1024}^2=2^{20}$ rather than 1,000,000 for the prefix Mega (M). If a wireless (WiFi) router transfers 150 Mbps of data, how many bits per second is that in decimal arithmetic?

(a) The ideal size (most efficient) for a broadcast antenna with one end on the ground is one-fourth the wavelength ($\lambda \text{/}4$) of the electromagnetic radiation being sent out. If a new radio station has such an antenna that is 50.0 m high, what frequency does it broadcast most efficiently? Is this in the AM or FM band? (b) Discuss the analogy of the fundamental resonant mode of an air column closed at one end to the resonance of currents on an antenna that is one-fourth their wavelength.

For red light of $\lambda =660\text{nm}$, what are f, $\omega$, and k?

(a) Two microwave frequencies authorized for use in microwave ovens are: 915 and 2450 MHz. Calculate the wavelength of each. (b) Which frequency would produce closer together hot spots in foods due to interference effects?

Distances in space are often quoted in units of light-years, the distance light travels in 1 year. (a) How many meters is a light-year? (b) How many meters is it to Andromeda, the nearest large galaxy, given that it is $2.54\times {10}^6$ ly away? (c) The most distant galaxy yet discovered is $13.4\times {10}^9$ ly away. How far is this in meters?

(a) What is the frequency of the 193-nm ultraviolet radiation used in laser eye surgery? (b) Assuming the accuracy with which this electromagnetic radiation can ablate (reshape) the cornea is directly proportional to wavelength, how much more accurate can this UV radiation be than the shortest visible wavelength of light?