EUS Tutoring Session Review Problem Set
Physics 158 - Midterm 2
Introductory Physics for Engineers II
(1) Imagine that the Earth were of uniform density and that a tunnel was drilled along the diameter from
the North Pole to the South Pole. Assume the Earth is a perfect sphere, and let R and M be the radius and
mass of the Earth, respectively.
(a) If an object were dropped into the tunnel, show that it will undergo simple harmonic motion.
(b) Find its period P of oscillation.
(c) Show that the period P of oscillation is equal to the period of a satellite orbiting Earth just at the surface.
Hint 1. Gauss’s Law for gravitational fields is
˛
X
g · dA = −4πG
Mi
i
It has an identical conceptual foundation to Gauss’s Law for electric fields. Note that g is the gravitational
field (which is analogous to the electric field E).
1
(2) A charged square insulating wire of side length a, of uniform linear charge density δ, is centred about
the origin of the xy plane with sides parallel to the x and y axes. A charge Q lies a distance l above its
centre. See the figure.
Find the magnitude and direction of the force F acting on the charge Q.
Q
l
a
2
(3) An AC generator produces 12 A rms at 400 V rms with power factor unity.
(a) Find the rms power produced by this generator.
(b) The generator voltage gets boosted by a step-up transformer to 12 kV. Find the power after the step-up
transformer, assuming no losses in the transformer.
(c) The power is then transmitted to an electrical load with wires having resistance 8 Ω each way, until it
reaches a step-down transformer. Determine the rms power loss in the wires.
(d) Determine the power available to the load.
3
(4) Consider an RLC circuit with L = 2.5 mH, R = 4 Ω, C = 500 µF, driven by an AC voltage source of
amplitude 24 V, and frequency 400 Hz.
(a) Find XL , XC , Z, and φ.
(b) Find the maximum current and maximum voltage across each of L, C, R.
(c) Find the time by which the driving voltage leads (or lags) the current.
(d) Find the maximum voltage across the combination R and C, and the time by which this voltage leads
(or lags) the current.
(e) Find the maximum voltage across the combination R and L, and the time by which this voltage leads
(or lags) the current.
(f) Find the maximum voltage across the combination L and C, and the time by which this voltage leads
(or lags) the current.
4
(5) Suppose that you have a series RLC circuit in an FM radio. You tune in on a broadcast using a
variable capacitor. There are two radio stations broadcasting: Station 1 broadcasts at ω1 = 6 · 108 , and
Station 2 broadcasts at ω2 = 5.99 · 108 . The inductance of the inductor is L = 10−6 H, and both stations
input the same voltage to the circuit.
(a) Find the value of the capacitor that you need in order to tune in to Station 1.
(b) Fix the value of C to be that found in part (a). If the mean power consumed by the circuit when tuned
to Station 1 is 100 times the mean power consumed by the circuit when tuned to Station 2, what is the value
of the resistor R?
5
(6) For a spherical distribution of charge ρ(r), consider the charge Q(r) enclosed by a concentric sphere
of radius r.
(a) Find dQ/dr.
(b) If Q(r) = 12e−2r , find ρ(r).
6
(7) A point charge q is at the origin. Consider a circular surface of radius a that is normal to k, at a
distance l from q. The centre of the circular surface is directly above the origin. See the image.
What is the electric flux through the surface?
l
q
7
(8) Consider an insulating sphere of radius R, centred at the origin with uniform charge density ρ. A
spherical cavity of radius a is scooped out, with centre at b = hb1 , b2 , b3 i, where a + |b| < R.
(a) Find the magnitude and direction of the electric field at any point within the cavity.
(b) Find the magnitude and direction of the electric field at the point c = Rb̂, where b̂ is the unit vector in
the b direction.
8
(9) Consider a ring of radius a = 10 m, centred at the origin of the xy plane. It is of uniform charge
density and has total charge Q. A charge q lies on the x axis at a distance l metres from the origin. See the
figure. Express answers in terms of q and Q (which are both measured in coulombs).
(a) Find the force on q due to the ring for l = 25 m > a.
(b) Find the force on q due to the ring for l = 5 m < a.
Q
q
9
Waves:
s
T
2π
v=
,k=
, P = 12 µω 2 A2 v, po = ρωvs0
µ
λ
r
Pav
v ± vL
γRT
0
I
,
β
=
10dB
log
,
I=
,
Doppler
Effect
f
=
f
v=
0
10 I0
M
4πr2
v ∓ vS
Beats: ∆f = f2 − f1 ,
y = A cos(kx ∓ ωt + φ)
Interference: k∆x + ∆φ = 2πn or π(2n + 1), n = 0, ±1, ±2, ±3, ±4, . . .
mv
mv
Standing Waves fm =
, m = 1, 2, 3, . . . , fm =
, m = 1, 3, 5, . . .
2L
4L
Constants:
1
C2
,
e = 1.6 × 10−19 C
k=
≈ 9 × 109 N m2 /C 2 ,
0 = 8.84 × 10−12
4π0
N m2
1
µ0 = 4π × 10−7 T m/A,
c= √
= 299, 792, 458m/s
0 µ0
Point Charge:
k|q|
kq
k|q1 q2 |
, |E| = 2 , V =
|F| =
+ Constant
r2
r
r
ˆ
ˆ
b
Electric potential and potential energy ∆V = Va − Vb =
dV
Ex = −
,
E = −∇V ,
dx
Maxwell’s Equations:
ˆ
ˆ
a
E · dl = −
a
E · dl
b
∆U = Ua − Ub = q(Va − Vb )
ˆ
Qenc
E · dA =
= 4πkQenc
0
S
dΦE
B · dl = µ0 (Ienclosed ) + 0
dt
C
Where S is a closed surface and C is a closed curve. ΦE =
B · dA = 0
S
ˆ
E · dl = −
ˆ
C
dΦE
dt
E · dA and ΦB =
ˆ
B · dA
Energy Density:
1 2
1
uE = 0 E 2 and uB =
B (energy per volume)
2
2µ0
Forces:
F = qE + qv × B, F = IL × B
Capacitors:
1 q2
0 A
q = CV , UC = · , For parallel plate capacitor with vacuum (air): C =
, Cdielectric = KCvacuum
2 C
d
Inductors:
1
2
EL = −L dI
dt , UL = 2 LI , where L = N ΦB /I and N is the number of turns.
For a solenoid B = µ0 nI where n is the number of turns per unit length.
DC Circuits: VR = IR, P = V I, P = I 2 R
(For RC circuits)q = ae−t/τ + b, τ = RC, a and b are constants
(For LR circuits)I = ae−t/τ + b, τ = L/R, a and b are constants
AC circuits: XL = ωL, XC = 1/(ωC), VC = XC I, VL = XL I
p
Imax
2
V = ZI, Z = (XL − XC )2 + R2 , Paverage = Irms
R, Irms = √
2
XL − XC
If V = V0 cos(ωt), then I = Imax cos(ωt − φ), where tan φ =
, Pav = Vrms Irms cos φ
R
µ0 Idl × r
Additional Equations: dB =
·
4π
r3
q
Rt
− 2L
R 2
1
LRC Oscillations: q = A0 e
cos(ωt + φ), where ω = ω02 − 2L
and ω02 = LC
10