Assignment 25: Radiation Energy and Momentum

Due: 8:00am on Friday, April 13, 2012

Note: To understand how points are awarded, read your instructor's Grading Policy.

Energy in Electromagnetic Waves

Electromagnetic waves transport energy. This problem shows you which parts of the energy are stored in the electric

and magnetic fields, respectively, and also makes a useful connection between the energy density of a plane

electromagnetic wave and the Poynting vector.

In this problem, we explore the properties of a plane electromagnetic wave traveling at the speed of light along the x

axis through vacuum. Its electric and magnetic field vectors are as follows:

.

Throughout, use these variables ( , , , , , , and ) in your answers. You will also need the permittivity of free

space and the permeability of free space .

Note: To indicate the square of a trigonometric function in your answer, use the notation sin(x)^2 NOT sin^2(x).

Part A

What is the instantaneous energy density in the electric field of the wave?

Hint A.1 Energy density in an electric field

Hint not displayed

Give your answer in terms of some or all of the variables in .

ANSWER:

= Correct

Part B

What is the instantaneous energy density in the magnetic field of the wave?

Hint B.1 Energy density in a magnetic field

Hint not displayed

Give your answer in terms of some or all of the variables in .

ANSWER:

= Correct

Part C

What is the average energy density in the electric field of the wave?

Hint C.1 Average value of

Hint not displayed

Give your answer in terms of and .

ANSWER:

= Correct

Part D

What is the average energy density in the magnetic field of the wave?

Hint D.1 Average value of

Hint not displayed

Give your answer in terms of and .

ANSWER:

= Correct

Part E

From the previous results, derive an expression for , the average energy density in the whole wave.

Hint E.1 Relationship among , , and

Hint not displayed

Hint E.2 Relationship between and for electromagnetic waves in vacuum

Hint not displayed

Hint E.3 Relationship among , and for electromagnetic waves in vacuum

Hint not displayed

Express the average energy density in terms of and only.

ANSWER:

= Correct

Part F

The Poynting vector gives the energy flux per unit area of electromagnetic waves. It is defined by the relation

.

Calculate the time-averaged Poynting vector of the wave considered in this problem.

Hint F.1 Relationship between and for electromagnetic waves in vacuum

Hint not displayed

Hint F.2 Relationship among , and for electromagnetic waves in vacuum

Hint not displayed

Give your answer in terms of , and and unit vectors , , and/or . Do not use or .

ANSWER:

= Correct

If you compare this expression for the time-averaged Poynting flux to the one obtained for the overall energy density, you find

the simple relation

.

Thus, the energy density of the electromagnetic field times the speed at which it moves gives the energy flux, which is a logical

result.

Magnetic Field and Poynting Flux in a Charging Capacitor

When a circular capacitor with radius and plate separation is charged up, the electric field , and hence the electric flux ,

between the plates changes. According to Ampère's law as extended by Maxwell, this

change in flux induces a magnetic field that can be found from

,

where is the permittivity of free space and is the permeability of free space. We can solve this equation to obtain the field

inside a capacitor:

,

where is the radial distance from the axis of the capacitor.

Part A

You might know already that it is possible to think of the energy stored in a charged capacitor as being stored in the

electric field between the plates. We will explore this idea by considering the flow of energy into the space between

the plates during the charging process. The capacitor is charged by a constant current , which flows for a time . At

the beginning of this charging process ( ), there is no charge on the plates.

The Poynting vector gives the flow of electromagnetic energy per unit area per unit time and is defined in terms of

the electric field vector and the magnetic field vector by the relation

.

Find an expression for the magnitude of the Poynting vector on the surface that connects the edges of the two

circular plates.

Hint A.1 Find the electric field

Hint not displayed

Express the magnitude of the Poynting vector in terms of , , , , , and other variables and parameters of the problem. Ignore

all fringing effects.

ANSWER:

= Correct

Part B

Calculate the total amount of energy that flows into the space between the capacitor plates from to , by first

integrating the Poynting vector over the surface that connects the edges of the two circular plates, and then integrating over time.

Hint B.1 How to approach the problem

Hint not displayed

Hint B.2 Integration over the surface

Hint not displayed

Hint B.3 Surface area of a cylinder

Hint not displayed

Hint B.4 Time integral

Hint not displayed

Hint B.5 A helpful integral

Hint not displayed

Express the total amount of energy in terms of , , , , , and other variables and parameters of the problem. Ignore all

fringing effects.

ANSWER:

=

Correct

Recalling that the capacitance of a parallel plate capacitor given in terms of the surface area of the plates and the distance

between the plates is

,

and also recalling that the charge on the plates at time is given by

,

we can see that we have expressed the energy stored by the capacitor in the familiar way,

,

even though we derived it in a different way using the Poynting vector.

Poynting Flux and Power Dissipation in a Resistor

When a steady current flows through a resistor, the resistor heats up. We say that "electrical energy is dissipated" by

the resistor, that is, converted into heat. But if energy is dissipated, where did it come from? Did it come from the

voltage source through the wires?

This problem will show you an alternative way to think about the flow of energy and will introduce a picture in which

the energy flows in many unexpected places--but not through the wires!

We will calculate the Poynting flux, the flow of electromagnetic energy, across the surface of the resistor. The

Poynting flux, or Poynting vector , has units of energy per unit area per unit time and is related to the electric field

vector and the magnetic field vector by the equation

,

where is the permeability of free space.

Consider a cylindrical resistor of radius , length , and resistance with a steady

current flowing along the axis of the cylinder.

Part A

Which of the following is the most accurate qualitative description of the the magnetic field vector inside the cylindrical

resistor?

ANSWER:

The magnetic field vector points radially away from the axis of the cylinder.

The magnetic field vector is everywhere tangential to circles centered on the axis of the cylinder.

The magnetic field vector points inward toward the axis of the cylinder.

The magnetic field vector points along the axis of the cylinder in the direction of the current.

Correct

Part B

Find the magnitude of the magnetic field inside the cylindrical resistor, where is the distance from the axis of the cylinder,

in terms of , , , , and other given variables. You will also need and . Ignore fringing effects at the ends of the cylinder.

Hint B.1 Ampère's law

Hint not displayed

Hint B.2 How to set up the integral

Hint not displayed

Hint B.3 Amount of current through a loop

Hint not displayed

ANSWER:

= Correct

Part C

What can you say about the electric field vector inside the resistor?

ANSWER:

The electric field vector points along the axis of the resistor in the direction of the current.

The electric field vector is zero inside the resistor and on its surface.

The electric field vector is confined to the surface of the resistor and points in the direction.

The electric field vector points radially outward--away from the axis of the cylinder.

The electric field vector is everywhere tangential to circles centered on the axis of the resistor that lie in the

plane perpendicular to the current direction.

Correct

Part D

What is the magnitude of the electric field vector ?

Hint D.1 Use Ohm's law

Hint not displayed

Hint D.2 Relationship between and

Hint not displayed

Give the magnitude of the electric field vector in terms of , , and other parameters of the problem.

ANSWER:

= Correct

Part E

In what direction does the Poynting vector point?

Hint E.1 Cross products in cylindrical coordinates

Hint not displayed

ANSWER:

The Poynting vector is zero inside the resistor including its surface.

Correct

Part F

Calculate , the magnitude of the Poynting vector at the surface of the resistor (not at the circular ends of the cylinder). To

answer this you need to take .

Hint F.1 Definition of the Poynting vector

Hint not displayed

Give your answer in terms of , , and other parameters of the problem.

ANSWER:

= Correct

Multiplying this value of the Poynting flux by the surface area of the resistor (which in this case is equivalent to integrating the

Poynting vector over the surface of the resistor), we recover the familiar expression for the power dissipated in a resistor

through which a current flows:

.

Exercise 32.22

A sinusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 36.6 and an electric-field amplitude of

5.00×10−2

at a distance of 350 from the antenna.

Part A

Calculate the frequency of the wave.

ANSWER:

=

8.20×108

Correct

Part B

Calculate the magnetic-field amplitude.

ANSWER:

=

1.67×10−10

Correct

Part C

Find the intensity of the wave.

ANSWER:

=

3.32×10−6

Correct

Satellite Television Transmission

A satellite in geostationary orbit is used to transmit data via electromagnetic radiation. The satellite is at a height of 35,000 km

above the surface of the earth, and we assume it has an isotropic power output of 1 kW (although, in practice, satellite antennas

transmit signals that are less powerful but more directional).

Part A

Reception devices pick up the variation in the electric field vector of the electromagnetic wave sent out by the satellite. Given

the satellite specifications listed in the problem introduction, what is the amplitude of the electric field vector of the satellite

broadcast as measured at the surface of the earth? Use for the permittivity of space and for

the speed of light.

Hint A.1 How to approach this problem

Hint not displayed

Hint A.2 Find the Poynting Vector

Hint not displayed

Hint A.3 Find the energy flux through a sphere

Hint not displayed

Express the amplitude of the electric field vector in microvolts per meter to three significant figures.

ANSWER:

=

7.00

Correct

Part B

Imagine that the satellite described in the problem introduction is used to transmit television signals. You have a

satellite TV receiver consisting of a circular dish of radius which focuses the electromagnetic energy incident from

the satellite onto a receiver which has a surface area of 5 .

How large does the radius of the dish have to be to achieve an electric field vector amplitude of 0.1 at the

receiver?

For simplicity, assume that your house is located directly beneath the satellite (i.e. the situation you calculated in the

first part), that the dish reflects all of the incident signal onto the receiver, and that there are no losses associated with

the reception process. The dish has a curvature, but the radius refers to the projection of the dish into the plane

perpendicular to the direction of the incoming signal.

Hint B.1 How to approach this problem

Hint not displayed

Hint B.2 The relationship between and

Hint not displayed

Give your answer in centimeters, to two significant figures.

ANSWER:

=

18

Correct

Radiation Pressure

A communications satellite orbiting the earth has solar panels that completely absorb all sunlight incident upon them. The total

area of the panels is .

Part A

The intensity of the sun's radiation incident upon the earth is about . Suppose this is the value for the intensity of

sunlight incident upon the satellite's solar panels. What is the total solar power absorbed by the panels?

Hint A.1 Definition of intensity

Hint not displayed

Express your answer numerically in kilowatts to two significant figures.

ANSWER:

=

14

Correct kW

Part B

What is the total force on the panels exerted by radiation pressure from the sunlight?

Hint B.1 Time derivative of a kinetic energy in relation to momentum

Hint not displayed

Hint B.2 Working out the power incident upon the panels

Hint not displayed

Hint B.3 Getting the units right

Hint not displayed

Express the total force numerically, to two significant figures, in units of newtons.

ANSWER:

=

4.70×10−5

Correct N

Problem 32.54

NASA is giving serious consideration to the concept of solar sailing. A solar sailcraft uses a large, low-mass sail and the energy

and momentum of sunlight for propulsion.

Part A

Should the sail be absorbing or reflective?

ANSWER:

Absorbing

Reflective

Correct

Part B

The total power output of the sun is . How large a sail is necessary to propel a -kg spacecraft against the

gravitational force of the sun?

Express your answer using two significant figures.

ANSWER:

=

6.5

Correct