Today’s agenda:Electromagnetic Waves.Energy Carried by Electromagnetic Waves.Momentum and Radiation Pressure of an Electromagnetic Wave.

Momentum and Radiation Pressure

EM waves carry linear momentum as well as energy.

The momentum density carried by an electromagnetic wave is

Today’s lecture is brought to you by the

letter P.

2Sdp =dV c

where dp is the momentum carried in the volume dV.

This equation is not on your equation sheet, but you have

permission to use it for tomorrow’s homework (if

needed) (32.26)

If we assume that EM radiation is incident on an object for a time t and that the radiation is entirely absorbed by the object, then the object gains energy U in time t.Maxwell showed that the momentum change of the object is then:

The direction of the momentum change of the object is in the direction of the incident radiation.

incident

Up = (total absorption)c

When the momentum carried by an electromagnetic wave is absorbed at a surface, pressure is exerted on that surface.

If instead of being totally absorbed the radiation is totally reflected by the object, and the reflection is along the incident path, then the magnitude of the momentum change of the object is twice that for total absorption.

The direction of the momentum change of the object is again in the direction of the incident radiation.

2 Up = (total reflection along incident path)c

incident

reflected

The radiation pressure on the object is the force per unit area:

From Newton’s 2nd Law (F = dp/dt) we have:For total absorption,

So

Radiation Pressure

FP= AF 1 dpP= =A A dt

Up= c

dU1 dp 1 d U 1 SdtP= = = =A dt A dt c c A c

incident

(Equations on this slide involve magnitudes of vector quantities.)

This is the instantaneous radiation pressure in the case of total absorption: S(t)P(t)= c

For the average radiation pressure, replace S by <S>=Savg=I:

averagerad

S IP = =c c

Electromagnetic waves also carry momentum through space with a momentum density of Saverage/c2=I/c2. This is not on your equation sheet but you have special permission to use it in tomorrow’s homework, if necessary. Today’s lecture is

brought to you by the letter P.

radIP = (total absorption)c

rad2IP = (total reflection)c

incident

incident

reflected

absorbed

Using the arguments above it can also be shown that:

If an electromagnetic wave does not strike a surface, it still carries momentum away from its emitter, and exerts Prad=I/c on the emitter.

Example: a satellite orbiting the earth has solar energy collection panels with a total area of 4.0 m2. If the sun’s radiation is incident perpendicular to the panels and is completely absorbed find the average solar power absorbed and the average force associated with the radiation pressure. The intensity (I or Saverage) of sunlight prior to passing through the earth’s atmosphere is 1.4 kW/m2.

3 2 32WPower=IA= 1.4 10 4.0 m =5.6 10 W=5.6 kWm

Assuming total absorption of the radiation:

32average -6

rad 8

W1.4 10S I mP = = = =4.7 10 Pamc c 3 10 s

-6 2 -52rad

NF=P A= 4.7 10 4.0 m =1.9 10 Nm

Caution! The letter P (or p) has been

used in this lecture for power,

pressure, and momentum!

That’s because today’s lecture is brought to you by the letter

P.

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New starting equations from this lecture:

0

1S= E B

2 2max max

average0 0

E cB1 1S = =2 c 2

max

max 0 0

E E 1= =c=B B

22

B E 00

1 1 Bu =u = E =2 2

22 max

0 max0

B1 1u = E =2 2

U 2 Up = or c c radI 2IP = or c c

, ,

2k= =2 f f = =ck

There are even more on your starting equation sheet; they are derived from the above!