Download RF Fundamentals

RF Fundamentals
FINAL EXAM
Due December 11
12:00 noon
Each problem is worth 0 points.
1. A sphere of radius 2 cm contains a volume charge density  given by
  4 cos 2 
(C / m 3 )
Find the total charge Q contained in the sphere. Hint:
d cos 
  sin  .
d
2. Three point charges, each with q = 3 nC, are located at the corners of a triangle in the x-y
plane, with one corner at the origin, another at (2 cm, 0, 0), and the third at (0, 2 cm, 0). Find
the force acting on the charge located at the origin.
3. In a given region of space, the vector magnetic potential is given by

A  xˆ 5 cos y  zˆ 2  sin x  Wb .
m

a. Determine β .

b. Use      A  dsˆ  Wb  to calculate the magnetic flux passing through a square loop




with 0.25-m-long edges if the loop is in the x-y plane, its center is at the origin, and its
edges are parallel to the x- and y-axes.
4. The electromagnetic generator shown in Fig. 6-12 is connected to an electric light bulb with a
resistance of 100 . If the loop area is 0.1 m2 and it rotates at 3600 revolutions per minute
in a uniform magnetic flux density,  o  0.2 T , determine the amplitude of the current
generated in the light bulb.
 m  in spherical coordinates, find the current crossing the spherical

5. Given J  10 3 sin  rˆ A
2
shell, r  0.02 m . Hint: sin 2  
1  cos 2
.
2

6. In cylindrical coordinates, β  2.0 / r ˆ T  . Determine the magnetic flux, , crossing the
dr
 ln r .
plane surface defined by 0.5m  r  2.5m and 0m  z  2.0m. Hint: 
r
z
2.0
dŝ

0
0.5
2.5
7. A line of positive charge (L = charge/unit length) is aligned along the y-axis as shown, what
is the electric field vector at the point P.
y
P
a
+L
x
-L
8. A standing wave consists of the coherent superposition of two electromagnetic waves. The
combination of a wave moving in the +x direction with one moving in the -x direction gives



rise to an electric field Ex, t   Em y sin( kx  t )  y sin( kx  t ).
a. Show that the electric field vanishes (for any time) whenever x  n / k . (These x
locations are known as nodes. Hint: Use trig. identities for sum and differences of an
angle.
b. What is the magnetic field for this wave?
c. Show that at time t = 0, the electromagnetic energy is completely contained in the electric
field, while at t =  (quarter period) the energy is completely contained in the
magnetic field.
9. What are the Maxwell equations, and how do they differ between electrostatics,
magnetostatics, and electrodynamics?
10. A square loop is coplanar with a long, straight wire carrying current
i(t )  2.5 cos 2  10 4 t
( A)
a. Determine the emf induced across a small gap created in the loop.
b. Determine the direction and magnitude of the current that would flow through a 4-
resistor connected across the gap. The loop has an internal resistance of 1-.
z
For a long wire carrying current,
recall that:
10 cm
10 cm
i(t)
l
x

 I
β  φˆ o
2r
5 cm
y