78387970 Chapter Wise Unit 4 Physics Particle Physics Questions
Physics 515A: ELECTROMAGNETIC THEORY Spring …physics.arizona.edu/~melia/515A.hwABC.pdf · Physics...
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Physics 515A: ELECTROMAGNETIC THEORY Spring 2018 Prof. Fulvio Melia Physics 447 (621-9651) [email protected] Problem A: Show that an arbitrary function f (x) can be expanded on the interval 0 ≤ x ≤ a in a modified Fourier-Bessel series f (x)= ∞ n=1 A n J ν y νn x a , where A n = 2 a 2 1 − ν 2 y νn 2 J ν 2 (y νn ) a 0 f (x) xJ ν y νn x a dx . Problem B: (a) Show that for a system of current-carrying elements in empty space, the total energy in the magnetic field is W = 1 2c 2 d 3 x d 3 x ′ J(x) · J(x’) |x − x’| , where J(x) is the current density. (b) If the current configuration consists of n circuits carrying currents I 1 , I 2 , I 3 ,..., I n , show that the energy can be expressed as W = 1 2 n i=1 L i I i 2 + n i=1 n j>i M ij I i I j . Exhibit integral expressions for the self-inductances (L i ) and the mutual inductances (M ij ). Problem C: Consider two current loops (as in Fig. 5.3 of Jackson) whose orientation in space is fixed, but whose relative separation can be changed. Let O 1 and O 2 be origins in the two loops, fixed relative to each loop, and x 1 and x 2 be coordinates of elements dl 1 and dl 2 , respectively, of the loops referred to the respective origins. Let R be the relative coordinate of the origins, directed from loop 2 to loop 1. (a) Starting from the expression for the force between the loops, show that it can be written F 12 = I 1 I 2 ∇ R M 12 (R) ,
Transcript of Physics 515A: ELECTROMAGNETIC THEORY Spring …physics.arizona.edu/~melia/515A.hwABC.pdf · Physics...
Physics 515A: ELECTROMAGNETIC THEORY
Spring 2018
Prof. Fulvio Melia
Physics 447 (621-9651)
Problem A: Show that an arbitrary function f(x) can be expanded on the interval 0 ≤
x ≤ a in a modified Fourier-Bessel series
f(x) =∞∑
n=1
AnJν
(
yνnx
a
)
,
where
An =2
a2(
1− ν2
yνn
2
)
Jν2(yνn)
∫ a
0
f(x)xJν
(
yνnx
a
)
dx .
Problem B: (a) Show that for a system of current-carrying elements in empty space, the
total energy in the magnetic field is
W =1
2c2
∫
d3x
∫
d3x′J(x) · J(x’)
|x− x’|,
where J(x) is the current density.
(b) If the current configuration consists of n circuits carrying currents I1, I2, I3,..., In,
show that the energy can be expressed as
W =1
2
n∑
i=1
Li Ii2 +
n∑
i=1
n∑
j>i
MijIiIj .
Exhibit integral expressions for the self-inductances (Li) and the mutual inductances (Mij).
Problem C: Consider two current loops (as in Fig. 5.3 of Jackson) whose orientation in
space is fixed, but whose relative separation can be changed. Let O1 and O2 be origins in
the two loops, fixed relative to each loop, and x1 and x2 be coordinates of elements dl1and dl2, respectively, of the loops referred to the respective origins. Let R be the relative
coordinate of the origins, directed from loop 2 to loop 1.
(a) Starting from the expression for the force between the loops, show that it can be written
F12 = I1I2~∇RM12(R) ,
where M12 is the mutual inductance of the loops,
M12(R) =1
c2
∮ ∮
dl1 · dl2|x1 − x2 +R|
,
and it is assumed that the orientation of the loops does not change with R.
(b) Show that the mutual inductance, viewed as a function of R, is a solution of the
Laplace equation,
∇2
RM12(R) = 0 .
The importance of this result is that the uniqueness of solutions of the Laplace equation
allows the exploitation of the properties of such solutions, provided a solution can be found
for a particular value of R.
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