Name:_______________________ ___
PHY2061
12-11-06
Exam 3
Closed book exam. A calculator is allowed, as is one 8.5×11” sheet of paper with your own
written notes. Please show all work leading to your answer to receive full credit. Numerical
answers should be calculated to at least 2 significant digits. Exam is worth 100 points, 25%
of your total grade.
UF Honor Code: “On my honor, I have neither given nor received unauthorized aid in doing
this exam.”
Sphere: S = 4π r 2
4
π = 3.1415927
V = π r3
3
1μ F = 10−6 F 1 pF = 10−12 F
e = 16022
.
× 10−19 C
g = 9.8 m/s 2
1 μ C = 10−6 C
1 eV = 1.6 × 10−19 J
1
K=
= 9 ×109 N m 2 / C 2
c = 3.0 × 108 m/s
ε 0 = 8.8542 ×10−12 C2 / N m 2
4πε 0
K μ
1
k = 2 = 0 = 10−7 T ⋅ m / A
μ0 = 4π k = 1.257 × 10−6 T ⋅ m /A
μ0ε 0 = 2
c
c
4π
qenc
q1q2
ρ
F
F = K 2 rˆ12
Φ E = v∫ Ε ⋅ dΑ =
∇⋅E =
E=
S
r
q0
ε0
ε0
U
V=
E = −∇V
W = −ΔU = ∫ F ⋅ ds
ΔV = − ∫ E ⋅ ds
C
C
q0
q
di
dq
1
1
1
= +
ΔVC =
ΔVL = L
i=
Ceff = C1 + C2
C
dt
dt
Ceff C1 C2
L
L
1
1
1
R=ρ
τ RC = RC
τ LR =
Reff = R1 + R2
= +
A
R
Reff R1 R2
ΔV = iR
P = Vi = i 2 R =
F = q(E + v × B )
μ i ds × r
dB = 0
4π r 3
τ = r×F
Φ B = ∫ B ⋅ dΑ
S
V2
R
U=
Q2
2C
U=
1 2
Li
2
ω = 2π f =
F = i L×B
v∫
C
B ⋅ ds = μ0 ienc
μ = NiA
ε = −N
Bwire =
μ0i
2π r
τ = μ×B
U = −μ ⋅ B
dΦB
dt
L=N
Page 1 of 10
ΦB
i
Barc =
μ0i
Φ
4π R
dBz
dz
2
2
ε0 E
B
u=
+
2
2 μ0
Fz = μ z
1
LC
Name:_______________________ ___
PHY2061
12-11-06
τ LR =
L
R
X L = ωL
S=
1
μ0
1
LC
ωLC =
XC =
1
ωC
E× B
ω = 2π f
I=
k=
2π
tan φ =
X L − XC
R
P
= Sav
A
εm
R2 + ( X L − X C )
2
i = im sin (ωt − φ )
ε = ε m sin ωt
I = I 0 cos 2 θ
λf =v
λ
im =
vn =
n1 sin θ1 = n2 sin θ 2
c
n
sin θ = m
λ
d
I = I 0 cos 2 θ
Φ E = v∫ E ⋅ dA =
S
v∫
C
qenc
ε0
ρ
ε0
C
S
∇⋅B = 0
∇×E = −
∂B
∂t
∇ × B = μ 0ε 0
⎛ ∂F ∂Fy ⎞ ⎛ ∂Fz ∂Fx ⎞
⎛ ∂Fy ∂Fx
∇ × F ≡ curl ( F ) = ⎜ z −
−
−
⎟ xˆ − ⎜
⎟ yˆ + ⎜
∂z ⎠ ⎝ ∂x
∂z ⎠
∂y
⎝ ∂y
⎝ ∂x
∂E ∂E y ∂Ez
∂
∂
∂
∇ = xˆ + yˆ
+ zˆ
∇ ⋅ E ≡ div E = x +
+
∂x
∂y
∂z
∂x
∂y
∂z
a ⋅ b = ax bx + a y by + az bz
π⎞
⎛
sin α = cos ⎜ α − ⎟
2⎠
⎝
E ⋅ ds = −
∂
B ⋅ dA
∂t ∫S
dΦE
d
= μ0ienc + μ0ε 0 ∫ E ⋅ dA
dt
dt S
B ⋅ ds = μ0ienc + μ0ε 0
∇⋅E =
v∫
Φ B = v∫ B ⋅ dA = 0
∂E
+ μ0 j
∂t
⎞
⎟ zˆ
⎠
a × b = ( a y bz − az by ) x − ( ax bz − az bx ) y + ( ax by − a y bx ) z
π⎞
⎛
cos α = − sin ⎜ α − ⎟
2⎠
⎝
Page 2 of 10
PHY2061
12-11-06
Name:_______________________ ___
1. For a certain driven series RLC circuit, the maximum generator EMF is 125V and the
maximum current is 3.20A. The current leads the generator EMF by 60º.
(a) [6 points] What is the impedance of the circuit?
(b) [6 points] What is the resistance of the circuit?
(c) [4 points] Is the circuit predominantly capacitive or inductive, and why?
Page 3 of 10
PHY2061
12-11-06
Name:_______________________ ___
2. [8 points] An alternating source drives a series RLC circuit with an EMF of maximum
amplitude 6.0V. The phase angle of the current is +30º. When the potential difference across
the capacitor reaches its maximum positive value of +5V, what is the potential difference
across the inductor (including sign)?
Page 4 of 10
PHY2061
12-11-06
Name:_______________________ ___
3. [8 points] In a certain region of space there are no magnetic fields present. An electric field
does exist, however. If the y-component of the electric field is E y = Em x , where Em is a
constant, what is the x-component of the electric field?
4. [8 points] The electric field between the plates of a parallel-plate capacitor whose plates
have a large circular radius R is given by E = Em sin ωt . What is the magnitude of the
magnetic field between the plates of the capacitor a distance r<R from the center?
Page 5 of 10
PHY2061
12-11-06
Name:_______________________ ___
5. The magnetic field of a plane electromagnetic wave propagating in vacuum is described by
B = Bm sin ( kx + ωt ) zˆ in SI units.
(a) [4 points] In what direction does the electromagnetic wave propagate?
(b) [6 points] Determine the expression for the electric field without introducing any new
parameters.
(c) [4 points] If ω = 4 × 1015 s-1 what is the wavelength of the electromagnetic wave?
Page 6 of 10
PHY2061
12-11-06
Name:_______________________ ___
6. [8 points] Sunlight reaching Earth has an average intensity of 1.2 kW/m2. Calculate the
maximum strength of the electric field assuming the sunlight is a plane wave.
7. [8 points] An unpolarized beam of light is sent into a stack of 4 polarizing sheets, oriented
so that the angle between the polarizing directions of adjacent sheets is 30°. What fraction of
the incident intensity is transmitted by the system?
Page 7 of 10
PHY2061
12-11-06
Name:_______________________ ___
8. [8 points] The heating of air near the surface of the road leads to the familiar phenomenon
of a mirage. Suppose the index of refraction of air well above the ground is 1.00027, but near
the ground there is a layer of air with an index of refraction of 1.00023. What is the
maximum angle from horizontal that will lead to total internal reflection of a light ray?
9. [6 points] A double slit interference experiment finds the third bright maximum from the
center, Θ=0, at an angle (measured from the center of the double slit to the viewing screen)
of 0.3o. If the wavelength of the monochromatic light used is 500 nm, what is the separation
of the slits?
Page 8 of 10
Name:_______________________ ___
PHY2061
12-11-06
10. [8 points] In the figure shown, assume the two light waves, of wavelength 555 nm in air,
are initially out of phase by 180º. The indices of refraction of the media are n1 = 1.46 and n2
= 1.72. What is the smallest value of L that will put the waves exactly in phase once they
pass through the two media?
n1
n2
Page 9 of 10
PHY2061
12-11-06
Name:_______________________ ___
11. [8 points] A thin anti-reflective coating with an index of refraction of n1 = 1.4 is placed
on a lens with an index of refraction of n2=1.5. What is the minimum coating thickness
needed to ensure that light of wavelength 490 nm and of perpendicular incidence will be
reflected from the two surfaces of the coating with fully destructive interference? Assume
that the lens+coating is in air.
Page 10 of 10