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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.013 Electromagnetics and Applications Problem Set #5
Fall Term 2005
Issued: 10/4/05
Due: 10/12/05
______________________________________________________________________________
Reading Assignment: Sections 1.3.2, 1.4, 1.6, 5.1, 5.3 Quiz 1 on Thursday, October 20 at 10-11 a.m. Will cover material through Problem Set #5. The Final Exam will be on Dec. 21, 1:30-4:30 p.m. Problem 5.1
For the following electric fields in a linear medium of constant dielectric permittivity ε and
magnetic permeability μ , find the free charge density ρ f , magnetic field H , and current density
J.
a) E=E 0 (x ix + y iy ) sin ω t
b) E=E 0 (y ix − x iy ) cos ω t
Problem 5.2
An electric field is of the form:
E=10Re ⎡e j(4π ×10
⎣
6
t − 4π ×10−2 z)
⎤ ix
⎦
volts/meter
a) What is the frequency f , wavelength λ , and speed of light in the medium?
b) If the medium has magnetic permeability μ 0 = 4π ×10−7 henry/meter, what are the
relative permittivity ε r , wave impedance η , and the magnetic field H ?
c) What is the Poynting vector, S=E×H?
2
Problem 5.3
x
K=i x K 0 cos ωt = i x K 0 Re[e jω t ]
ε 0 ,μ 0
y
z
•
plasma
ε,μ 0
∂J
= ω2p εE
∂t
A sheet of surface current with the surface current density:
K=i x K 0 cos ωt = i x K 0 Re[e jω t ]
amperes/meter
is located at z = 0 with free space extending from −∞ < z < 0 and a plasma extending for z > 0 .
The plasma region has dielectric permittivity ε and magnetic permeability μ 0 of free space with
current constitutive law:
∂J
= ω2p
εE
∂t
a) If all fields are of the form: J(z,t)=Re[J(z)e jω t ]
l
E(z,t)=Re[E(z)e jω t ]
l
H(z,t)=Re[H(z)e jω t ]
what is the plasma complex conductivity, σ (ω ) , in the plasma region defined as:
l
J(z)=σ (ω ) E(z) ?
3
b) Ampere’s law in the plasma region becomes:
l l
l
∇ × H = J(z) + jωεE = jωε(ω)E .
What is the frequency dependent permittivity ε(ω) ?
l
c) Assume that E(z) is of the form:
⎧⎪Eˆ p e-jk p z ix
l
E(z)= ⎨
jk z
⎪⎩Eˆ 0 e 0 ix
z>0
z < 0
What are the wavenumbers k p and k 0?
l
d) What is the general form of H(z) in each region in terms of Eˆ 0 and Eˆ p ?
e) What are the boundary conditions at z=0 on the electric and magnetic fields?
ˆ
ˆ
l0 , E
l p , E(z),
f) Solve for E
and H(z).
g) Find the time average Poynting vector, < S >=
evaluate for ω > ωp and ω < ωp .
4
1 ⎡ ˆ ˆ *⎤
Re E × H ⎥ , for z < 0 and z > 0 and
⎦
2 ⎢⎣
Problem 5.4
n1
n2
(1)
S 0
45˚
(2)
Se
A glass prism in the shape of an isosceles right triangle has an index of refraction n1 . The
surrounding environment has index of refraction n 2 . Neglect multiple internal reflections within
the prism.
a) What is the minimum value of n1 for no time average power to be transmitted across the
prism hypotenuse when the prism is in free space (n 2 = 1) or in water (n 2 = 1.33) ?
b) If the incident light at point (1) has time average power per unit area S0 , what is the
exiting time average power Se at point (2) in terms of the refractive index ratio n=n1/n2
assuming that n1 is above the minimum value for no time average power to be transmitted
across the prism hypotenuse. Evaluate S0/Se for the two minimum values of n1 found in
part (a) for free space or water surrounding the prism.
5