Physics 212 – Midterm Exam II                               NAME:_Albert Einstein_________

Summer 2003                                                               STUDENT ID#:_______________

Prof. Milos Steinhart

 

INSTRUCTIONS: This is a closed book exam. You must show all of your work in the space provided for each question and you must include appropriate units and significant figures. If you need more space, use the other side of the paper but mark clearly to which problem and which question is your work related. Try to make your work as clear and transparent as you can. Even if you understand the problem well, think twice, what you are precisely asked.

Do not use other papers!

You are bound by the University’s Honor Code. Any evidence of cheating must be reported to the College of Arts and Sciences.

 

 

 

 

PLEASE CIRCLE ALL ANSWERS

All problems are worth 20 points.

 

 

 

 

 

 

 

 

SCORE

                                                           

                                                            Problem 1 ___20________________

 

                                                            Problem 2 ___20________________

 

                                                            Problem 3 ___20________________

 

                                                            Problem 4 ___20________________

 

                                                            Problem 5 ___20________________

 

                                                            TOTAL    __100________________

Problem 1: A galvanometer has an internal resistance of Rg = 30 W and deflects full scale for an If = 50 mA current. Its accuracy is 1%. Make drawings and describe how to use this galvanometer to make:

 

a) (10) An ammeter to read currents up to Ia = 30 A ?

 

From Ohm’s law the voltage on the galvanometer at full deflection is

Vf = 1.5 mV.

If we build an ammeter we must use a resistor Ra in parallel to the galvanometer to divert the excess current around.

Since the current through this resistor Ia >> Ig = 50 mA, we can expect that the resistor Ra has (almost) all the current through it. So at full-scale deflection Ia = 30 A and Va = Vg = 1.5 mV Þ

Ra = Va/Ia = 1.5 mV/30 A = 5 10-5 W.

To get all the precision we would have to use Ia - Ig in the denominator but since the accuracy is 1% we can neglect Ig in comparison with Ia.

 

 

 

 

 

 

 

 

 

b) (10) A voltmeter to give a full-scale deflection of Vv = 1000 V ?

 

If we build a voltmeter we must use a resistor Rv in series to the galvanometer so it has the rest of the total voltage on it.

Since the voltage across this resistor Vv >> Vg = 1.50 mV, we can expect that the resistor Rv has (almost) all the voltage across it. So at full-scale deflection Iv = 50 mA and Vv = 1000 V Þ

Rv = Vv/Iv = 1000 V/50 mA = 20 MW.

To get all the precision we would have to use Vv - Vg in the numerator but since the accuracy is 1% we can neglect Vg in comparison with Vv.

 

 

 

 

 

 

 

 

Problem 2: A coaxial cable consists of a solid inner conductor of radius R1, surrounded by a concentric cylindrical tube of inner radius R2 and outer radius R3. The conductors carry equal and opposite currents I0 distributed uniformly across their cross-sections.

Determine magnetic field at a distance r from the axis for:

 

 

a) (5) r < R1

 

The current densities in the conductors are:

Jinner = I0/pR12 and Jouter = I0/p(R32 - R22) (with the opposite directions).

Because of the cylindrical symmetry, we use in all cases the Ampere’s law. The closed path will always be a circular with radius r.

 

2prB = m0pr2Jinner = m0I0pr2/pR12 Þ B = r m0I0/2pR12.

The magnetic induction grows linearly.

 

 

b) (5) R1 < r < R2

 

2prB = m0I0 Þ B = m0I0/2pr.

Whole I0 is surrounded, the magnetic induction decreases as 1/r.

 

 

 

 

 

 

 

c) (5) R2 < r < R3

 

2prB = m0I0 - m0I0p(r2 – R22)/p(R32 – R22) Þ

B = m0I0 /2pr [1 - (r2 – R22)/(R32 – R22)] = m0I0 /2pr [(R32 - r2)/(R32 – R22)].

Superposition of the whole I0 and a part of - I0.

 

 

 

 

 

 

d) (5) r > R3

 

Now both currents which are the same and opposite are fully surrounded.

2prB = m0(I0 - I0) = 0 Þ B = 0.

 

Problem 3: A beam of electrons flies from the left into the magnetic field perpendicularly to its field lines, which emerge perpendicularly from the plane of the figure and have a magnitude of B = 4.55 10-4 T. The radius of the orbit is r = 3.52 cm.

 

a) (4) Make a drawing of the orbit and mark clearly how exactly the electrons move, including all the directions and orientations.

 

The electrons fly counterclockwise as would any negatively charged particles.

 

 

 

 

 

 

 

b) (6) What is the speed v of the circulating electrons? Are relativistic effects significant at this speed?

 

From the equilibrium of the eccentric and Lorenz forces

mv2/r = qvB Þ

v = qBr/m = 1.6 10-19 C 4.55 10-4 T 3.52 10-2 m/9.11 10-31 kg = 2.81 106 m/s,

which is not yet relativistic so our classical formula is OK.

 

 

 

 

c) (5) What is the period T of revolution of the circulating electrons?

 

T is time of one turn, distance of the path over speed Þ

T = 2pr/v = 6.28 3.52 10-2 m / 2.81 106 m/s = 7.87 10-8 s = 78.7 ns.

 

 

 

 

 

 

d) (5) What voltage V had to be used to accelerate the electrons from zero to their current speed?

 

The kinetic energy of the electrons is gained at the cost of a lost of their potential energy mv2/2 = qV Þ

V = mv2/2q = 9.11 10-31 kg (2.81 106) m2s-2 /2 1.6 10-19 C = 22.48 V.

The energy can be written as 22.48 eV!

Problem 4: An elastic circular conductive loop lies in a plane of drawing, which is perpendicular to uniform magnetic field. The vector of magnetic induction has a direction out of the plane of the drawing and the magnitude B0 = 0.48 T. The area of the loop changes at a constant rate dA/dt = -3.5 10-2 m2/s. At t = 0 the loop has area A = 0.285 m2.

 

a) (4) Make a drawing and mark clearly what will be the direction of the current caused by the EMF induced by the change.

 

Since the rate of change of the area is negative the area shrinks and also the magnetic flux decreases. The induced EMF will cause a current, which will produce magnetic field in the same direction as has the original field to reinforce it and thereby oppose the change. So the current is counterclockwise.

 

 

 

 

b) (6) Determine the induced EMF at time t = 0 s?

 

EMF = -dFm/dt = -BdA/dt = - 0.48 T(- 3.5 10 –2 m2/s) = 16.8 mV

 

 

 

 

 

 

 

 

c) (6) Determine the induced EMF at time t = 2.00 s?

 

The EMF will be again   16.8 mV since it depends on the rate of the change of the flux rather than on its absolute value.

 

 

 

 

 

d) (4) Let’s cut the loop at one point. What will be the polarity of the EMF on the sides of the cut? Mark it clearly in you drawing or make a new drawing and explain it!

 

The cut loop must behave like a power source, which, when the gap is closed, produces a counterclockwise current. When one comes to the gap in the counterclockwise direction he first comes to the positive pole. One can also see how charge moves in every little piece of the loop.

Problem 5: Multiple choice. Pick the single best answer to each conceptual question below. Please include a brief explanation. Each answer is worth of 4 credits.

 

1. A current flows in a long straight wire along the x-axis in the positive x-direction. The magnetic field due to this current in a point P on the negative y-axis points to which direction:

a)      –x

b)      +x

c)      –y

d)      +y

e)      –z

f)        +z

 

2. A small planar current loop is placed in a uniform magnetic field. The magnitude of the torque on the loop in a maximum when:

a)      the plane of the loop is parallel to the direction of the field.

b)      the plane of the loop is perpendicular to the direction of the field.

c)      the angle between the plane of the loop and the direction of the magnetic field is somewhere between 0 and 90°.

d)      the torque is independent of the angle between the plane of the loop and the direction of the magnetic field.

 

3. The magnetic energy stored in an inductor is

a)      proportional to the square of the current through the inductor.

b)      proportional to the square of the magnetic field in the inductor.

c)      both of the above.

d)      neither of the above.

 

4. A resistor R and an inductor L are connected in series to a battery, which is switched on at t = 0. The current in the circuit is time-dependent. If we repeat the experiment with a resistor of resistance 5R, the time constant

a)      decreases by the a factor of 5.

b)      increases by the a factor of 5.

c)      does not change.

 

5. A resistor and an initially uncharged capacitor arranged in series are charged by a battery, which is connected at time t = 0. The current in the circuit

a)      is constant because the EMF supplied by the battery is constant.

b)      decreases exponentially in time.

c)      increases exponentially in time.

d)      There is no current because the electrons cannot flow through the gap in the capacitor.