−27
kg
kg
T · m/A
Fx
~
F
θ
Fy
Circular Motion
s
∆θ
θ=
ω=
r
∆t
2πr
vt =
vt = ωr
τ
2π
v2
ac = t
τ =
r
ω
~ | sin θ
F y = |F
q
~ | = Fx2 + Fy2
|F
Fy
θ = arctan
Fx
~ = Fx x̂ + Fy ŷ
F
~ | cos θ
F x = |F
Vectors
2
vf2 s = vis
+ 2as ∆s
sf = si + vis ∆t + 21 as (∆t)2
vf s = vis + as ∆t
Kinematics
µ0 = 4π × 10
−7
c = 3.00 · 108 m/s
g = 9.81 m/s
2
mp = 1.67 · 10
me = 9.11 · 10
−31
0 = 8.85 · 10−12 C2 /Nm2
qe = 1.6 · 10−19 C
k = 8.99 · 109 Nm2 /C2
Physical Constants
∆p
∆t
∆U
∆t
=
E
Req + r
1
1
+
+ ...
=
R1
R2
E −t/(RC)
e
R
q(t) = ECe−t/(RC)
Discharging RC circuit
E
i(t) = e−t/(RC)
R
q(t) = EC(1 − e−t/(RC) )
Charging RC circuit
RC circuits
R, C, L Circuits
i(t) =
L
A
= R1 + R2 + R3 + . . .
parallel
Req
1
series
Req
P = IV
Vterm = E − Ir, I =
ρ (Ω · m)
1.47 × 10−8
1.72 × 10−8
2.44 × 10−8
20 × 10−8
= IR, R = ρ
1
1
+
+ ...
C1
C2
Circuits
∆Q
I=
,V
∆t
Material
Silver
Copper
Gold
Steel
series
Ceq
1
Capacitance
Q
C=
V
A
C = 0 (parallel plate)
d
parallel
Ceq
= C1 + C2 + . . .
nb
na
nb
Brewster Angle
na
1
1
1
+ 0 =
s
s
f
1
1
1
= (n − 1)
+
f
R1
R2
Thin Lens Equation
Geometric Optics
y0
s0
m≡
=m=−
y
s
1
1
1
+ 0 =
s
s
f
nb − na
na
nb
+ 0 =
s
s
R
na s0
m=−
nb s
tan θp =
θc = arcsin
λ = λ0 /n
na sin θa = nb sin θb
n = c/v
Refraction
ΦB = BA cos φ
N
I
L
n = N/L (Number density)
X
Bk ∆s = µ0 Ienc
Bsolenoid = µ0 nI = µ0
µ0 I
2π r
on wire
Bwire =
FB
v2
R
= I`B sin φ
Fc = |q|vB = m
direction given by RHR
FB = |q|vB sin φ
The Magnetic Field
∆i1
∆t
∆i
∆t
uB =
B2
2µ0
uE = 12 0 E 2
Energy field density (J/m3 )
Uinductor = 12 LI 2
E2
N2
=
E1
N1
Transformers
E =L
AN1 N2
l
Self Inductance
M = µ0
A is area of smaller coil
M for 2 cocentric coils
E2 = M
N2 |ΦB2 | = M |i1 |
Mutual Inductance
E(t) = ωAB sin (ωt)
∆ΦB
∆t
Generator
E = −N
Inductance
∆ΦB
E =−
∆t
Coil with N turns
1
0 µ0
Common Prefixes
femto
f
10−15
pico
p
10−12
nano
n
10−9
micro
µ
10−6
milli
m 10−3
centi
c
10−2
kilo
k
103
mega
M 106
giga
G 109
tera
T 1012
A = 2πrL + 2(πr2 )
Cylinder: V = πr2 L
Sphere: V = 43 πr3 , A = 4πr2
Circle: A = πr2 , C = 2πr
Useful Geometry
P = I/c (radiation pressure)
It = 12 Ii (if Ii is unpolarized)
It = Ii cos2 φ
1 ∆U
S=
= 0 cE 2
A ∆t
2
I = Savg = 12 0 cEmax
2π
, ω = 2πf, τ = 1/f
λ
u = 0 E 2
k=
B(t, x) = Bmax sin (ωt ∓ kx)
E(t, x) = Emax sin (ωt ∓ kx)
v = λf, E = cB, c = √
EM Waves
Print Name:
• Read each question carefully.
RL Circuits
Uq+sources = qVsources
E U = 12 CV 2
i(t) =
1 − e−t(R/L)
R
r
V = V0 + Ex (parallel plate)
1
LC Circuits : ω =
1 q
LC
V =
(point charge)
4π0 r
Electric Potential
q1 q2
Uq1 q2 = k
r
Uparallel plate = U0 + qEx
ΦE = EA cos θ
~ = k q radially outward
E
r2
~ //plate = Q = V
E
0 A
d
∆V
E=−
∆s
Gauss’s Law
Qenc
ΦE =
0
Electric Forces & Fields
|q1 q2 |
FE = k 2
r
~ net = E
~1 + E
~2 + E
~3 + . . .
E
~on q = q E
~
F
Power: P =
W = F d cos θ
Wi→f = Uf − Ui
Ki + Ui = Kf + Uf
Emech = K + U
K = 12 mv 2 , Ug = mgh
Energy
p
~ = m~v , Faverage =
~net = F
~1 + F
~2 + F
~3 + . . .
F
~A on B = −F
~B on A
F
Newton’s Laws
~net = m~a
F
Old Dominion University
PHYS232 Final-Exam
2021-Fall
UIN:
• Clearly mark your answers and show your work.
• Clearly explain your answers.
• Include units for all answers.
• Attach your notesheet and any work at the end of test for 5pts.
• You may use a calculator. Do not share calculators.
• Cheating of any kind is grounds to receive a zero on the entire test.
Honor Pledge: I pledge to support the Honor System of Old Dominion University. I will refrain from any
form of academic dishonesty or deception, such as cheating or plagiarism. I am aware that as a member of
the academic community it is my responsibility to turn in all suspected violations of the Honor Code. I will
report to a hearing if summoned.
Signature:
1/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
1. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 1
Suppose scientist announce the discovery of a brand new sub-atomic particle in physics. They named it the
Coulton because of its unique property that two Coulton particles electric repulsion is equal in magnitude to
their gravitational attraction towards each other.
(a) If a Coulton has the same electric charge of a proton, how much mass does a Coulton have?
(b) How many times more massive is a Coulton than a proton?
Givens: qp = 1.602 × 10−19 C, mp = 1.673 × 10−27 kg, G = 6.674 × 10−11 Nm2 /kg2 .
2/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
2. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 2
Four charges are fixed in position as shown. The electric field at the center points . . . ? Draw a vector sum of
the four electric fields to support your answer.
q2
q1
q2
q1
Givens: q1 = −3.80 nC,
q2 = 4.00 nC
3/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
3. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 3
In order to make a nuclear bomb using Uranium you must first process the raw material. This means separating
isotopes 238U (mass of 3.95 × 10−25 kg) and 235U (mass of 3.90 × 10−25 kg). You construct a tunnel with a length
L and a uniform electric field of strength E, pointing from the floor to the ceiling. You fire the Uranium atoms
into your device with a speed v. Treat the isotopes as having the same amount of charge, Q = 1.47 × 10−17 C.
Q: How strong must E be so the vertical separation between
238
U and
235
U at the end is h?
Hint: Assume no magnetics fields and use the kinematic equations.
Givens: L = 27.00 m, v = 7.50 × 105 m/s, h = 27.00 cm
4/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
4. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 4 Charge Q0 = −3.33 nC is at the origin, and
charge Q1 = 3.79 nC is initially at point A (1,0). How much work is necessary to move charge Q1 from point
A to point B (9,-8)? (all position measurements are in meters)
0
Q0 A
−2
−4
−6
B
−8
0
2
4
6
8
10
5/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
5. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 5 Each diagram below depicts the equipotential
lines around the given charge distribution.
Draw the electric fields for each diagram.
+
+
−
+
−
−
−
−
6/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
6. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 6
The equivalent capacitance of this circuit is . . .
C1
C2
C3
C4
= 80 µF
= 30 µF
= 90 µF
= 50 µF
7/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
7. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 7
Set up and solve the equations required to determine the currents in the follow circuit. Clearly label the
equations you create and designate if you used a KVL or KJL to get the equation.
8/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
8. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 8
Consider two (infinitely long) parallel current carrying wires separated by d = 51.7 cm. I1 = 4.60 A, and
I2 = 5.43 A
(a) Draw the magnetic field above and below wire 1.
(b) Draw the magnetic field above and below wire 2.
(c) In which region can you find a space where the net magnetic field is ~0T?
A. Above wire 2.
B. Between wire 1 and wire 2.
C. Below wire 1.
D. There is no region where the net B-field cancels.
(d) If it exist, find the distance above or below the x-axis where the net magnetic field is ~0T.
9/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
9. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 9
A single-turn rectangular coil has a resistance of 2.0Ω. The magnetic field at all points inside the coil varies
according to B = B0 e−αt , where B0 =0.25T and α=200Hz.
What is the current induced in the coil at (a) t=0.001s, (b) 0.002 s, (c) 2.0 s?
10/ 11
Old Dominion University
PHYS232 Final-Exam
2021-Fall
10. (10 points) Nerem PHYS232 2021 Fall Final Exam Problem 10
Light rays fall normally on the vertical surface of the glass prism (nprism =1.50).
(a) What is the largest value for φ such that the ray is totally reflected at the slanted face?
(b) Repeat the calculation of part (a) if the prism is immersed in water (nwater =1.3).
11/ 11