Name:_______________________ ___
PHY2061
9-28-06
Exam 1
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 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.”
4
π = 3.1415927
e = 16022
.
× 10−19 C
V = π r3
3
a ⋅ b = ax bx + a y by + az bz
a × b = ( a y bz − by az ) x − ( ax bz − bx az ) y + ( ax by − bx a y ) z
Sphere: S = 4π r 2
1
K=
4πε 0
qq
F = K 1 2 2 rˆ12
r
E = −∇V
∇ = xˆ
∫
V
ε 0 = 8.8542 ×10−12 C2 / N m 2
= 9 ×109 N m 2 / C 2
E=
V=
U
q0
∂
∂
∂
+ yˆ + zˆ
∂x
∂y
∂z
F
q0
Φ E = v∫ Ε ⋅ dΑ =
S
K=
1 2
mv
2
∇ ⋅ F = div ( F ) =
1
Q2
2
U = C ( ΔV ) =
2
2C
1μ F = 10−6 F
x=
ε0
−b ± b 2 − 4ac
2a
ρ
ε0
ΔV = − ∫ E ⋅ d s
C
S
1 μ C = 10−6 C
∇⋅E =
W = −ΔU = ∫ F ⋅ ds
∇ ⋅ F dV = v∫ F ⋅ dΑ
Q = C ΔV
qenc
c = 3.0 × 108 m/s
C
∂Fx ∂Fy ∂Fz
+
+
∂x
∂y
∂z
Reff = R1 + R2
1
1
1
= +
Reff R1 R2
Ceff = C1 + C2
1
1
1
= +
Ceff C1 C2
1 pF = 10−12 F
1 eV = 1.6 ×10−19 J
g = 9.8 m/s 2
Page 1 of 10
R=ρ
L
A
i=
dq
dt
ΔV = iR
Name:_______________________ ___
PHY2061
9-28-06
y
+4q
+2q
−3q
−2q
−2q
x
s
+2q
+2q
1. A central particle of charge −3q is surrounded by a hexagonal array of other charged
particles (q>0). The length of a side is s, and charges are placed at each corner.
(a) [6 points] Find the component of the force along the x-axis (Fx) on the central
particle.
(b) [6 points] Find the component of the force along the y-axis (Fy) on the central
particle.
Page 2 of 10
Name:_______________________ ___
PHY2061
9-28-06
y
+λ
−λ
R
x
2. Consider electric charge distributed along a one-dimensional path in the form shown
as two sections of ¼ of a circle each. The circle is centered at the origin with a radius
of R, and the linear charge density is +λ in the left quadrant and −λ in the right.
(a) [6 points] Find the component of the electric field along the x-axis (Ex) at the
origin (0,0).
(b) [6 points] Find the component of the electric field along the y-axis (Ey) at the
origin (0,0).
Page 3 of 10
Name:_______________________ ___
PHY2061
9-28-06
y
x
z
3. Consider a cube with side length s = 2 m and one corner at the origin (0,0,0) as
shown.
(a) [6 points] What is the total charge enclosed by the cube if the electric field is
E = 3x 2 ˆi + 5ˆj − 2 z kˆ N/C ?
(
)
(b) [6 points] What is the electric charge density (C/m3) at the center of the right
face at x = 2 m if the electric field is the same as in part (a)?
Page 4 of 10
Name:_______________________ ___
PHY2061
9-28-06
z
R
4. [8 points] A flat nonconducting surface infinite in extent carries a uniform charge
density of σ = 3 × 10−9 C/m 2 . A small circular hole of radius R = 1.5 m has been cut
in the middle of the sheet as shown. Calculate the electric field at a point z = 5 m
away from the center of the hole along an axis perpendicular to the surface. (In other
words, consider z R , but don’t set R / z exactly equal to zero. You may find the
superposition principle useful.)
Page 5 of 10
Name:_______________________ ___
PHY2061
9-28-06
v
+
+
+
+
+
+
+
5. [8 points] An electron is launched away from the surface of an infinite
nonconducting sheet of charge with a velocity of v = 2 ×106 m/s on a trajectory
perpendicular to the surface. The charge density of the sheet is σ = +5 nC/m2. Is the
electron able to reach a distance infinitely far away from the charged sheet, and if not,
how far does it travel before turning around? The charge of the electron is
q = −e = −1.6 × 10−19 C , and the electron mass is me = 9.11×10−31 kg .
Page 6 of 10
PHY2061
9-28-06
Name:_______________________ ___
6. A conducting sphere of radius R1 contains a charge Q. It is surrounded by a
concentric spherical conducting shell of radius R2 > R1 and charge −Q.
(a) [6 points] What is the difference in electric potential between the shell and
the sphere?
(b) [6 points] What is the capacitance of the arrangement of conductors?
Page 7 of 10
PHY2061
9-28-06
Name:_______________________ ___
7. [6 points] Initially two electrons are fixed in place with a separation of 2.15 µm. How
much work must be done to bring a third electron in from infinity to complete an
equilateral triangle?
8. [6 points] Sketch the electric field lines for a negatively charged particle above the
surface of a flat perfectly conducting surface (both above and below the surface).
−
conductor
Page 8 of 10
PHY2061
9-28-06
Name:_______________________ ___
9. A 1.5 µF capacitor is charged to a potential difference of 12 V, and the charging
battery is disconnected.
(a) [6 points] What is the energy stored in the capacitor?
(b) [6 points] If the charged capacitor is then connected in parallel with a second
(initially uncharged) capacitor, and if the potential difference across the first capacitor
subsequently drops to 9 V, what is the capacitance of this second capacitor?
Page 9 of 10
PHY2061
9-28-06
Name:_______________________ ___
10. [6 points] The electric potential along the x-axis (in V) is plotted versus the value of
x, (in cm). Evaluate the x-component of the electrical force (in Newtons, including
sign) on a proton located on the x-axis at x = 10 cm.
11. [6 points] Two wires are made out of the same material (copper). One has a circular
cross section with radius r = 1 mm and a length of 10 cm, the other has a square cross
section with width s = 1 mm and a length of 5 cm. Which wire has the larger
resistance?
Page 10 of 10