Download exam i, physics 1306

FINAL EXAM, PHYSICS 2401, May 10, 2013, Dr. Charles W. Myles
INSTRUCTIONS: Please read ALL of these before doing anything else!!!
1.
2.
3.
4.
5.
PLEASE put your name on every sheet of paper you use and write on one side of the paper only!!
PLEASE DO NOT write on the exam sheets, there will not be room! This wastes paper, but it makes
my grading easier!
PLEASE show all work, writing the essential steps in the solutions. Write formulas first, then put in
numbers. Partial credit will be LIBERAL, provided that essential work is shown. Organized, logical,
easy to follow work will receive more credit than disorganized work.
The setup (PHYSICS) of a problem will count more heavily than the math of working it out.
PLEASE write neatly. Before handing in your solutions, PLEASE: a) number the pages & put the
pages in numerical order, b) put the problem solutions in numerical order, & c) clearly mark your final
answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves.
NOTE!! The words “EXPLAIN”, “DISCUSS” & “DEFINE” below mean to answer mostly in
ENGLISH, NOT math symbols!
Constants that may be needed: Permittivity of free space: ε0 = 8.85  10-12 C2/(N m2).
k = [1/(4πε0) = 9.0  109 N m2/C2]. Permeability of free space: μ0 = 4π  10-7 T m/A
NOTE: I have 14 exams to grade! PLEASE HELP ME GRADE
THEM EFFICIENTLY BY FOLLOWING THE ABOVE SIMPLE
INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY
RESULT IN A LOWER GRADE!! THANK YOU!!
Three 8.5’’ x 11’’ pieces of paper with anything on them & a calculator are allowed. Question 1
(Conceptual) is REQUIRED! NOTE: You MUST ALSO WORK Problem 2 (Electromagnetism)!! Also
work any three (3) of the remaining problems for five (5) problems total on the exam. Each problem is
equally weighted & worth 20 points, for a total of 100 points. Answer all parts of the ones you choose!
1. MANDATORY CONCEPTUAL QUESTIONS!!! Answer briefly all parts in a few
complete, grammatically correct English sentences. I want answers which use mainly ENGLISH
WORDS, NOT symbols or equations! If you insist on using symbols, DEFINE all symbols you use!
(Note: Answers with ONLY symbols, with no WORD explanations get ZERO credit!).
a.
b.
c.
d.
e.
f.
g.
h.
i.
State Coulomb’s Law.
State Gauss’s Law.
State Ampère’s Law.
State the Biot-Savart Law.
State Faraday’s Law of Induction.
State Lenz’s Law.
State Gauss’s Law for Magnetism.
State Maxwell’s Equations of Electromagnetism.
Briefly Define the following terms: i) Electric Field, ii) Electric Potential, iii) Magnetic Field.
j.
iv) Electromotive Force (emf).
Briefly Define the following terms: i) Electric Flux, ii) Magnetic Flux iii) Displacement Current,
iv) Electromagnetic Wave.
NOTE!!! Parts k & l are BONUS QUESTIONS, 5 POINTS EACH!
k. To arrive at the relations known as Maxwell’s Equations, Maxwell made a modification
to Ampère’s Law so that it’s form resembled (but wasn’t exactly the same as!)
Faraday’s Law with the electric field E & the magnetic field B interchanged & the
electric flux ΦE & the magnetic flux ΦB interchanged. Briefly tell me what modification
Maxwell made to Ampère’s Law to achieve this result.
l. When Maxwell’s Equations are combined to show that Electromagnetic Waves are
produced by time varying fields, it is found that, in vacuum, the velocity of these waves c
is exactly the speed of light in vacuum. In this derivation, it is shown that c can be
expressed in terms of two fundamental constants that are used in earlier chapters: The
permittivity of free space ε0 & the permeability of free space. Write the equation which
relates c to ε0 & μ0.
2. MANDATORY PROBLEM ON ELECTROMAGNETISM!!! Fig. 1 shows a parallel
plate capacitor with circular plates of radius r0 & area A = π(r0)2 = 100 cm2. There is air
between the plates. The capacitance C = 30 pF (30  10-12 F). It is being charged by a V0 =
70 V battery through a resistor of R = 2.0 Ω (NOT SHOWN in the figure!). At the instant the
battery is connected (time t = 0), the initial charge on the plates is zero.
Fig. 1
It is at that same instant when the electric field between the plates is
changing most rapidly. As time goes on, the charge Q on the plates
increases with time towards a maximum value Qmax. Assume that the
V0
electric field E between the plates is spatially uniform and that it is zero
anywhere outside the edges of the plates. Calculate:
a. The maximum charge Qmax which will be on the plates after a very
long time has passed.
b. The time constant τ for this RC circuit.
c. The current I flowing into the plates at t = 0.
For the following, recall that the formula for the time dependent charge Q(t) on the
capacitor plates in an RC circuit is Q(t) = CV0[1 – exp(-t/τ)]. Also recall that
the electric field between the plates of a parallel plate capacitor is E = (σ/0 ),
Fig. 2
where σ is the surface charge density (Q/A) on the plates.
d. Write a formula for the time rate of change (dE/dt) of the electric field
between the plates. Calculate the numerical value of this quantity at t = 0.
e. Write a formula for the Maxwell Displacement Current between the
plates. Calculate the numerical value of this displacement current at t = 0.
f. Assuming that the field lines associated with the magnetic field B induced
between the plates are circles perpendicular to the electric field E, as in
Fig. 2, write a formula for the time dependence of this magnetic field at the outside edge
(at r = r0) of the plates. Calculate
g. the numerical value of this field at t = 0.
NOTE: WORK ANY THREE (3) OF PROBLEMS 3., 4., 5., & 6!!!!!
3. Figure 3 shows a non-conducting sphere of radius r0, which contains a spherical cavity of
radius r1 (r1 < r0) centered at the same point as the larger sphere. A positive charge Q is
uniformly distributed throughout the volume between the spheres (between r1 & r0).
Calculate:
a. The electric field everywhere outside the outer sphere (r > r0).
b. The electric field everywhere inside the inner sphere (0 < r < r1).
c. The electric field everywhere between the inner & outer spheres (r1 < r < r0).
d. The electric potential everywhere outside the outer sphere. (r > r0)
e. The electric potential everywhere inside the inner sphere (0 < r < r1).
f. The electric potential everywhere between the inner & outer spheres (r1 < r < r0).
The following may be needed: Differential volume element in spherical coordinates (with
no angular dependence): dV = 4πr2dr. Integral of r to a power ∫rndr = [rn+1/(n+1)] (n ≠ -1).
NOTE: WORK ANY THREE (3) OF PROBLEMS 3., 4., 5., & 6!!!!!
4.
Answer parts a & b briefly using correct English, NOT with symbols!
a. State Kirchhoff’s Junction Rule for electrical circuits. What
Physical Principle is the basis for this Rule?
b. State Kirchhoff’s Loop Rule for electrical circuits. What Physical
Principle is the basis for this Rule?
For the electrical circuit in Fig. 4, Calculate (in any order!)
c. The current I1.
d. The current I2.
e. The current I3.
f. The power dissipated in the 30 Ω resistor.
Fig. 4
Fig. 3
NOTE: WORK ANY THREE (3) OF PROBLEMS 3., 4., 5., & 6!!!!!
5. Figure 5 shows a circular cross section of a length ℓ of a cylindrical
ℓ
coaxial cable. The inner conductor is a thin, hollow, conducting tube
of radius r1 & the outer conductor is a thin, hollow conducting tube of
radius r2. The conductors carry equal currents I in opposite
directions. The spaces inside the inner conducting tube (r < r1) &
between the conducting tubes (r1 < r < r2) are filled with air.
Calculate the magnetic field B:
a. At all points inside the inner conductor (r  r1).
b. At all points between the two conductors (r1  r  r2).
c. At all points outside the outer conductor (r2  r).
Answer part d briefly using correct English, NOT math symbols!
d. What Physical Principle did you use to calculate the magnetic fields in parts a, b & c?
For the length ℓ of the coaxial cable, Calculate:
e. The total magnetic flux passing through it.
f. The self-inductance.
Fig. 5
NOTE: WORK ANY THREE (3) OF PROBLEMS 3., 4., 5., & 6!!!!!
6. Figure 6 shows a conducting bar of length ℓ = 0.5 m moving to
the right on two frictionless rails. A uniform magnetic field
directed into the page has magnitude B = 0.4 T. The resistor has
resistance R = 10.0 Ω. The conducting bar is moving to the right
at a constant velocity v = 12 m/s. Calculate:
a. The induced (motional) emf Ɛ in the rectangular loop formed
by the rails, the conducting bar, and the resistor.
  x 
b. The induced current I produced by the emf Ɛ in the loop.
What is the direction of this current?
c. The power delivered to the resistor.
Since the induced current I in the bar is in the magnetic field B, the bar is subject to a
magnetic force to the left. In order for the bar to move to the right at a constant velocity,
an external force must be applied to the right to balance this left-directed magnetic force.
d. Calculate the magnitude of this magnetic force that the conducting bar experiences.
e. What Physical Principle did you use to answer part a?
Fig. 6