Second Summer Session 2010
Physics 222 Exam #2
The examination is closed book and will begin at 10 am and end at 12 noon. Some potentially useful
formulas are provided on a separate sheet. Each problem is worth 20 points. You must show all your
work!
1. A non-conducting sphere of radius 2 cm has a total charge
of 1.5 µC spread uniformly throughout its volume. A metallic,
spherical shell of inner radius 4 cm and outer radius 6 cm is
concentric with this non-conducting sphere and has a total
charge of –3 µC on it. Using Gauss’ Law,
(a) Find the electric field E for r=1 cm.
(b) Find the electric field for r=3 cm.
(c) Find the electric field for r=5 cm.
(d) Find the electric field for r=10 cm.
(e) Find the charge on the inner surface of the metal shell
and on the outer surface of this shell.
2. A point charge Q is located a distance d above a nonconducting plane of charge density Q0/A where A is the area
of the plane. Both Q and Q0 are positive. Treating the plane
as an infinite plane, find the magnitude and direction of the
total electric field
(a) halfway between the plane and the charge Q (point Pa).
(b) a distance d above the plane and a distance d from the
charge Q (at point P)
(c) What choices for Q0 (in terms of Q, A, and d) will make
the total electric field equal zero at some point?
3.
(a) What is the electric potential at the surface of a non-conducting sphere of charge Q radius R?
If a charge q of mass m is brought from infinity to this surface
(b) how much work is done by the field?
(c) how much has the potential energy of the field changed?
If the charge q is then released
(d) what is the instantaneous force on it? (magnitude and direction)
(e) what is its final (maximum) speed?
4. One capacitor is charged until its stored energy is 4.0J. A second uncharged capacitor is then connected
to it in parallel.
(a) If the charge distributes equally, what is now the total energy stored in the electric fields?
(b) Where did the excess energy go?
(c) If the capacitance of the first capacitor is 1 µF, what was the initial voltage across the first capacitor?
(d) What is the final voltage across the two capacitors in parallel?
5. Wires A and B, having equal lengths of 40.0 m and equal diameters of 2.60 mm, are connected in series.
A potential difference of 60.0 V is applied between the ends of the composite wire. The resistances of the
wires are 0.127 Ω and 0.729 Ω, respectively. Determine
(a) the current density in each wire (Hints: use charge conservation; the total voltage drop across both wires
is the sum of the voltage drop across A and that across B)
(b) the potential difference across each wire
(c) the electric field magnitude in each wire
Formulas
1 |q1 ||q2 |
4π²0 r2
F =
1
= 8.99 × 109 N · m2 /C 2
4π²0
~
~ = F
E
q0
e = 1.6 × 10−19 C
1 |q|
4π²0 r2
I
~
~ · dA
Φ= E
λ
2π²0 r
E=
1 q
E=
4π²0 r2
Winf ty
q
f
Vf − Vi = −
U
q
~ · d~s
E
V =
i=1
Vi =
1
4π²0
n
X
qi
i=1
Es = −
ri
∂V
∂s
Ex = −
E=−
q = CV
C = 4π²0
1
=
Ceq
n
X
ab
b−a
1
Cj
µ
σ
²0
q
4π²0 R3
¶
W
q
Uf
Ui
∆U
∆V = Vf − Vi =
−
=
q
q
q
Z f
1 q
~ · d~s
V =−
E
V =
4π²0 r
i
i
n
X
σ
2²0
E=
U = −W∞
V =
Z
E=
E=
E=0
∆U = Uf − Ui = −W
V =−
~
F~ = q E
E=
²0 Φ = qenc
1atm = 1.013 × 105 P a
C=
²0 A
d
∆V = Vf − Vi = −
∂V
;
∂x
Ey = −
∂V
;
∂y
∆V
∆s
L
ln(b/a)
n
X
Ceq =
Cj
C = 2π²0
C = 4π²0 R
j=1
2
q
= 12 CV 2
u = 12 ²0 E 2
2C
j=1
Z
dq
V
~
i=
i = J~ · dA
R=
dt
i
1
E
ρL
~ = ρJ~
ρ= =
E
R=
σ
J
A
V2
ρ − ρ0 = ρ0 α(T − T0 )
P = iV
P = i2 R =
R
U=
1
Ez = −
∂V
∂z