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PHY 3323
September 28, 2009
Exam #1
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Thucydides V.99
(1) Evaluate the following integrals:
R
a) 26dx (3x2 −2x−1)δ(x−3) (7 points).
R
b) 05dx cos(x)δ(x−π) (7 points).
R
c) 03dx x3 δ(x+1) (7 points).
R∞
d) −∞
dx ln(x+3)δ(x+2) (7 points).
(2) Compute the line integral of ~v = r cos2 (θ)b
r − r cos(θ) sin(θ)θb + 3r φb around the 3-step
path:
a) Along the x axis from the origin to x
b (7 points);
b) Counterclockwise along the unit circle of radius 1 in the xy plane from x
b to yb (7
points); and
c) Along the y axis from yb back to the origin (7 points).
d) Check your answer using Stokes’ theorem (7 points).
(3) Suppose the charge density is given in cylindrical coordinates (s, φ, z) as
n
for 0 ≤ s ≤ R .
ρ = ks
0
for R < s
a) What are the dimensions of the constant k? (11 points)
~ everywhere in space? (11 points)
b) What is the electric field E
c) What is the potential V everywhere in space? Take the origin as your reference
point (11 points).
(4) Suppose the potential in spherical coordinates (r, θ, φ) is V = Ae−λr .
a) What are the dimensions of the constants A and λ? (9 points)
~ everywhere in space? (9 points)
b) What is the electric field E
c) What is the charge density ρ everywhere in space? (9 points)
d) What is the electrostatic energy W of this system? (9 points)