Name:_______________________ ___
H-ITT or pseudonym: ______________
PHY2061
10-21-04
Exam 2
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.
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.”
4
V = π r2
3
Sphere: S = 4π r 2
1
K=
π = 3.1415927
me = 9.11×10−31 kg
ε 0 = 8.8542 ×10−12 C2 / N m 2
= 9 ×109 N m 2 / C 2
4πε 0
q
ρ
qq
Φ = v∫ Ε ⋅ dΑ = enc ∇ ⋅ E =
F = K 1 2 2 rˆ12
S
ε0
ε0
r
U
F
V=
E=
W = −∆U = ∫ F ⋅ ds
C
q0
q0
∂
∂
∂
∇ = xˆ + yˆ + zˆ
∂F ∂Fy ∂Fz
∂x
∂y
∂z
∇ ⋅ F = div ( F ) = x +
+
∂x
∂y
∂z
F = grad ( f ) = ∇f
Q = C ∆V
1
Q2
2
U = C ( ∆V ) =
2
2C
∆V = iR
P = Vi = i 2 R =
R=ρ
L
A
i=
c = 3.0 × 108 m/s
1
γ =
1 − v 2 / c2
u x′ =
ux ± v
vu
1 ± 2x
c
p = γmu
k=
E = −∇V
∆V = − ∫ E ⋅ d s
C
∫
V
∇ ⋅ F dV = v∫ F ⋅ dΑ
S
Ceff = C1 + C2
1
1
1
= +
Ceff C1 C2
Reff = R1 + R2
1
1
1
= +
Reff R1 R2
dq
dt
1 eV = 16022
.
× 10−19 J
t = γ t0
u y′ =
F = dp / dt
F = q (E + v × B )
V2
R
e = 16022
.
× 10−19 C
L=
L0
x′ = γ ( x ± vt )
t ′ = γ ( t ± vx / c
γ
uy
E = γmc 2
⎛ vu x ⎞
γ ⎜1 ± 2 ⎟
c ⎠
⎝
2 4
m c = E 2 − p2c2
F = i L×B
K µ0
=
= 10−7 T ⋅ m / A
2
c
4π
B wire =
2
)
K = ( γ − 1) mc 2
µ0i
2ki
rˆ =
rˆ
r
2π r
µ0 = 4π k = 1.257 × 10−6 T ⋅ m /A
Page 1 of 10
y′ = y
z′ = z
Name:_______________________ ___
H-ITT or pseudonym: ______________
PHY2061
10-21-04
R1
ε
a
+
R2
R3
b
1. Consider the circuit above. The battery supplies an EMF of ε = 6.0 V, and the resistances
of the 3 resistors are R1 = 100Ω , R2 = 25Ω , and R3 = 75Ω
(a) [6 points] Find the voltage drop between points a and b: ∆Vab = Va − Vb .
(b) [6 points] Find the current through resistor R2.
Page 2 of 10
PHY2061
10-21-04
Name:_______________________ ___
H-ITT or pseudonym: ______________
(c) [6 points] Find the total power dissipated in the circuit that must be provided by the
battery.
(d) [8 points] Suppose an identical circuit (with the same resistors and battery
configuration) is connected to the above circuit at points a and b. That is, a conductor
connects point a of the first circuit to point a of the second circuit, and another
conductor connects point b of the first circuit to point b of the second. Calculate the
voltage drop between points a and b for this combined circuit. (Hint: apply Kirchoff’s
laws to each loop of the circuit.)
Page 3 of 10
Name:_______________________ ___
H-ITT or pseudonym: ______________
PHY2061
10-21-04
R1
a
+
ε
C1
C2
S
b
3. Consider the circuit above. A battery supplies an EMF of ε = 12.0 V, the resistance
R1 = 1.25 MΩ , and the capacitors have capacitance C1 = 18µ F and C1 = 9µ F . Consider that
switch S is closed at time t = 0. ( 1 MΩ = 106 Ω , 1µ F = 10-6 F ).
(a) [6 points] What is the time constant of the circuit that governs how fast the current
changes after the switch is closed ?
(b) [6 points] What is the maximum charge attained on the top plate of capacitor C1
(i.e. at point a) after the switch is closed?
Page 4 of 10
PHY2061
10-21-04
Name:_______________________ ___
H-ITT or pseudonym: ______________
(c) [6 points] Sketch the voltage drop ∆Vab = Va − Vb across the capacitors as a function of
time t. Try to be accurate in your sketch.
Page 5 of 10
PHY2061
10-21-04
Name:_______________________ ___
H-ITT or pseudonym: ______________
3. A new type of cannon can launch 5 kg projectiles at the high velocity of 0.3c
(a) [6 points] What is the minimum amount of energy required to accelerate a projectile
from rest to this final velocity?
(b) [6 points] If the cannon is mounted on the front of a spaceship traveling away from
Earth at a velocity of 0.6c, what will be the velocity of the launched projectile in the
reference frame of the Earth?
Page 6 of 10
PHY2061
10-21-04
Name:_______________________ ___
H-ITT or pseudonym: ______________
4. A kaon is an unstable particle with a rest mass energy of 500 MeV and an average
lifetime of τ = 5.2 ×10−8 s in its rest frame. Consider a beam of kaons, where each kaon
has a momentum of 1000 MeV/c as measured in a laboratory. (1 MeV = 1.6 ×10−13 J )
(a) [6 points] What is the velocity of each kaon in the laboratory?
(b) [6 points] What is the average distance the kaons travel at this momentum in the
laboratory before decaying?
Page 7 of 10
PHY2061
10-21-04
Name:_______________________ ___
H-ITT or pseudonym: ______________
5. A long straight wire has an electric charge density of λ = 7.0µ C / m along its length,
where 1µ C = 10−6 C , as measured in the rest frame of the wire.
(a) [6 points] What is the observed charge density of the wire as measured by an
observer traveling at a velocity of v = 2.4 ×108 m/s parallel to the direction of the wire?
(b) [6 points] What is the observed charge density of the wire as measured by an
observer traveling at a velocity of v = 2.4 ×108 m/s perpendicular to the direction of the
wire?
Page 8 of 10
Name:_______________________ ___
H-ITT or pseudonym: ______________
PHY2061
10-21-04
y
− − − − − −
E
x
e-
d
+ + + + + + +
6. [8 points] A beam of electrons (“cathode rays”) is sent between two parallel electric plates
separated by a distance d = 1.5 cm with a potential difference of 300 V between them. The
electric field points in the + yˆ direction. If the electron beam travels perpendicular to the
electric field in the +xˆ direction, and each electron has a kinetic energy of 5,000 eV, what
magnetic field is necessary (direction and magnitude) so that the electrons continue traveling
in a straight line without deflection by the electric field? The charge of the electron is
q = −e = −1.6 × 10−19 C , the electron rest-mass energy is 511,000 eV, and
1 eV = 1.6 ×10−19 J .
Page 9 of 10
Name:_______________________ ___
H-ITT or pseudonym: ______________
PHY2061
10-21-04
iA
d
C
iB
7. Consider two parallel wires, A and B, separated by a distance d = 0.5 cm as shown above.
The current of wire A is iA = 9 A , and the current of wire B is iB = 3 A , both in the
x̂ direction.
(a) [6 points] Calculate the direction and magnitude of the magnetic field at point C,
midway between the two wires in the plane of the wires.
(b) [6 points] Find the direction and magnitude of the force per unit length (per meter)
on wire B arising from the interaction of its current with the magnetic field of wire A.
Page 10 of 10