Download Discussion Question 13B

Discussion Question 13B
P212, Week 13
Electromagnetic Waves
A laser beam travels through vacuum. The electric field of the plane electromagnetic wave
produced by the laser has the form given below. The wavelength of the beam is λ = 514 nm, and
the amplitude of the electric field is E0 = 2.5 x 104 N/C.
E (x, y,z,t) = yˆ E 0 cos(kz + ωt − 45 )
(a) In what direction is this wave propagating?
What does “propagating wave” mean anyway? It means that the shape of the wave remains
the same as a function of time, it just moves. The shape of the wave is determined by that
cosine. As time rolls forward, how do you have to change position so that the shape looks
the same?
G
ˆ 0 cos(−450 ). This spot obeys
Lets follow a spot on the wave corresponding to E = yE
kz + ωt = 0 or z = -
ω
ω
t. Hence this spot moves along the -zˆ axis and evidently = c
k
k
(b) What are the magnitudes of the wave number k and angular frequency ω?
8
2π
2π
2πc 2π ( 3 ×10 )
7
-1
k=
=
= 1.22 × 10 m ; ω = ck =
=
= 3.66 × 1015 Hz
514 × 10−9
λ 514 × 10−9
λ
(c) Make a sketch of what the E field looks like at some moment in time.
π⎞
⎛
At t = 0, we have E y = E0 cos ⎜ kz − ⎟
4⎠
⎝
which is a cosine curve that maximizes
π
at z = +
4k
Ey
t =0
z
(d) Write down an expression for the magnetic field B(x,y,z,t). Express your answer
algebraically (i.e. no numbers) in terms of the symbols k, ω, and B0 (the latter being the
magnetic field amplitude). Be sure to indicate the direction of the magnetic field.
The E and B fields of an electromagnetic wave have a fixed relationship to the direction of
propagation of the wave. You know the direction of the wave, and the direction of E, so you
can figure out the direction of B with the help of your sketch. Add the B-field to your
drawing in part (c).
The B field has the same position and time
dependence as the E field. It is directed
G G
such that E × B is along the direction of
propagation (marked S in the figure)
G
B ( x, y, z, t ) = xˆ B0 cos(kz + ωt − 45D )
E
z
y
x
B
S
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E (x, y,z,t) = xˆ E 0 cos(kz + ωt − 45 )
where E0 = 2.5 x 104 N/C and λ = 514 nm
(e) What is the amplitude B0 of the magnetic field?
E0 2.5 × 104
B0 =
=
= 8.33 × 10−5 T
8
c
3 ×10
(f) What is the intensity I of the wave? Be sure to indicate the units of your answer.
2
2.5 × 104 )
(
Erms
I =
=
= 8.29 × 105 W/m 2 or alternatively
Z0
2( 377 Ω)
2
−5
4
Erms Brms Emax Bmax ( 2.5 × 10 )( 8.33 × 10 )
I =
=
=
= 8.29 × 105 W/m 2
−7
µ0
2µ 0
2 ( 4π×10 )
(g) A solar cell is an example of a photo-sensitive detector that absorbs the energy of
incident electromagnetic radiation and converts it into useful power (with which to operate
your solar-cell calculator, for example). Suppose our electromagnetic wave is incident on a
small, square photo-sensitive detector of side 2 mm. You can assume that this detector is
smaller than the wave itself, and is oriented so that its surface is perpendicular to the
incoming beam. You can also assume that the detector is 100% efficient at converting all
the radiative energy incident on it into useful power. What, then, is the average power <P>
that our detector would generate?
As always, units are an enormous help. The units of your answer to part (e) tell you how to
determine the answer to this question!
detector
The intensity is the power per unit area.
P = I A = ( 8.29 × 105 W/m 2 )( 0.0022 m 2 ) = 3.32 W
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