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Physics 2306

Fall 2010 Exam 3

November 12, 2010 1. Write your name and student number on the ‘Answer’ sheet. 2. There are 20 equivalent problems. You MUST show units where appropriate

for it to be considered correct. Note that some problems are MUCH easier than others. I will not give partial credit this time – so be sure to make sure your answers are correct and complete in detail.

3. Assume SI units in problems unless otherwise indicated. 4. During the exam you can use a calculator, but not its programmable features. 5. You will have two hours to work on the exam. 6. The exam is closed book and closed notes. Only the formula sheets provided

may be used. THE LAST FOUR FORMULA SHEETS CONTAIN OUTRIGHT METHODOLOGIES FOR EACH PROBLEM – USE THEM! Be sure to transfer all your answers (including units) to the ‘Answer’ sheet; other pages will not be graded.

7. By writing your name on this exam, you pledge that you have upheld the Honor

Code and have arrived at each answer solely by your own work. 8. Do not share this exam with others until after the make-up exam on Monday. For problems where you are asked to rank items, items of different magnitudes should be separated by commas, while those of the same magnitude should have no comma between them. For example:

Rank the following numbers from highest to lowest A) 7 B) 5 C) 5 D) 3 Answer: A,BC,D

If there are any questions on these instructions, be sure to ask. Answers I can not read will not be given credit. Be careful to not put in extra commas by accident.

(for those taking make-up exam – note that answer sheet is the last page – remove it and turn in only that one page)

1) Two inductors are in parallel as shown, and are then in series with a resistor and a battery. What is the current in the resistor long after the switch is closed?

2) Two long parallel wires are perpendicular to the paper as shown. The first caries 2A going into the paper, and the second 1A also into the paper. Sketch the value of By on the x-axis from a large negative x to a large positive x.

3) A loop of wire in the x-y plane encloses an area A, and is placed in a uniform magnetic field whose initial values and rates of change are as shown. Rank the current in the loop in the direction indicated, from most positive to most negative in the four cases indicated.

4) A straight segment of wire runs from (0,0,0) to (x,y,z) and carries a current I. It is located in a uniform magnetic field given by B = (Bx, By, Bz) What is the total force on the wire in the z direction?

5) A rectangular loop of wire is positioned near a long straight wire as shown. What is the magnitude of the magnetic flux through the loop due to a current I in the straight wire?

6) Electric and magnetic fields exist in a region of space with directions shown. A positively charged particle is projected at speed v in directions A, B, or C. Rank these directions by which one would give the greatest magnitude initial force on the particle. (All the vectors in the diagram are in the same plane.)

7) Four resistors, an uncharged capacitor and an inductor are wired as shown and

with the values indicated. Rank the three currents, A, B, C from the largest to the least, just after the switch is closed.

8) In terms of kg, m, s, C, what are the units of a Tesla?

9) A standard co-axial cable (like for cable TV) has the dimensions shown. Find its capacitance per unit length. (Reduce to simplest form only – do not plug in numbers.)

a = radius of centre core = 5.0 x 10-4 m b = inner radius of metallic shield = 2.3 x 10-3 m (dielectric constant) = 2.3

10) Find the inductance per unit length for the cable in problem 9. (Reduce to

simplest form only – do not plug in numbers.)

11) Two vertical wires, separated by a distance l, are placed in a uniform magnetic field of magnitude B in the z direction as shown. The top of the wires are connected by a resistor R, while a slider of mass m completes the circuit and slides down the two wires under the force of gravity. The slider and wires have negligible resistance. What is its terminal speed v?

12) Find the power dissipated in the 4 resistor in the following circuit. (remember to include units!)

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13) The capacitor is initially uncharged. The switch is then closed. What is the charge on the capacitor when the system comes to equilibrium?

14) What must the resistance R be such that the voltage at A is 4 V, assuming it is

zero at the negative terminal of the battery?

15) What is the power going into the magnetic field of the inductor 3.0 µs after the switch is closed?

16) A long conducting rod, of radius 2 cm, has a current of 10 A flowing uniformly along it. What is the magnitude of the magnetic field at a radius of 1 cm?

17) What is the differential equation describing the displacement x for the following

oscillator in terms of the given parameters. Write it in the form 0

(ie: replacing 2 for this situation).







20) What is the differential equation describing the charge q for the following



In an attempt to have the class average come out naturally at 81.5%, I offer the following suggestions for each of the problems (and possibly useful equations):

1) What is the voltage drop across the inductors (or equivalent inductor) a long time after the switch is closed?

2) Pick several representative points along the x-axis, and at each draw a vector indicating the relative size (and direction) of the B field due to each of the wires. B-fields then add like vectors, sketch a smooth curve in the answer sheet based on these results.

3) The current flows in the direction which would result in a reduced change of magnetic flux within the loop.

4) Consider how one can find the cross-product using the determinant of a matrix.

5) Find the B field a distance r from the wire (using Ampere’s Law with a loop around the wire). Then use that to calculate the flux in the rectangular loop, noting that you’ll have to integrate from r=a to r=b.

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6) Find the direction of the force due to the electric field, and then that due to the magnetic field (due to the particle’s speed v. The net force is the vector sum of the two.

7) Consider carefully what will happen just after the switch closes. Which elements will have voltage across them at that point?

8) Consider the Lorentz force.

9) A charge per unit length along the central conductor will produce an electric field between the two conductors. You can get the electric field using Gauss’ Law and a cylinder co-axial with the central conductor as your ‘Gaussian’ surface. The integral of E ‘dot’ dr will then result in a voltage between the two. Then consider the definition of capacitance: Q = CV. [or C/L = (Q/L)/V for capacitance per unit length] The Q should cancel out, allowing you to solve for C (per unit length). Remember that the dielectric constant reduces the magnitude of E.

10) A current down the central conductor will produce a magnetic field around it. You can find this using Ampere’s Law. (What about the region outside the outer conductor? – the return current is in the outer conductor) Then you can find the flux through a rectangular loop of length l running along the cable, with one side touching the inner conductor, and the opposite side the inner edge of the outer conductor. A change in the flux through this loop, due to a change in current in the central conductor, will induce an EMF. Then consider the definition of inductance, and you should be able to extract the inductance per unit length.

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11) As the slider moves down, a current will be induced through the loop. Consider the direction of the force on the slider due to this current. When it is equal and opposite the force of gravity, there will be no further acceleration. You can also do this by energy conservation (gravitational potential energy being converted completely into heating the resistor – and no longer going into kinetic energy).

12) Find the equivalent resistance of the parallel resistors, then the voltage at the top of the 2 ohm resistor. This gives you the voltage across the 4 ohm resistor, which should let you determine its power.

13) Consider carefully what happens a LONG time after the switch is closed. Where is current still flowing? That should let you get the voltage across the capacitor, and thus the charge on the capacitor.

14) The current which flows through the upper resistor is split between the two lower ones. You can figure out how much goes through the 5 ohm one by the conditions stated in the problem. This means you can then solve for R.

15) By finding the current flowing through the circuit after 3 microseconds, you can figure out the voltage across the inductor, and recall that P = VI.

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16) Since the current density is uniform throughout the rod, you can draw an “Amperian” loop at the desired radius, and figure out how much current flows within that loop.

17) I’ve done this one enough times in class so it should not be a problem.

18) Remember that the restoring force is the component of the gravitational force along the ‘x’ direction (which is distance along the arc-length). For small angles, sin(theta) ~ x/L.

19) Same as 18, but now there is another force besides gravity, and so mass will not simply cancel out.

20) Draw a loop and apply Kirchoff’s Law. Then recall that I is already the first time derivative of the charge: dq/dt.

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Name (PRINT):__________________________________ Student Number #: ______________________________

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ANSWER SHEET (Exam 3, 2306 Fall 2010) First 3 letters of last name: (be clear!)

the last four formula sheets contain outright

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