e+m qualifier

8/13/2019 E+M Qualifier

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Part IIElectricity and Magnetism with Optics

21 January 2 11In this three-hour examination you are required to answer all twenty multiple-ehoicequestions and to solve either Problem 1 or Problem 2, and either Problem r Problem 4. should take you about 1.25 hours for the multiple-choice questions and about 1.75hours for the two problems. Use one answer booklet for the answers to the multiplechoice questions, and one answer booklet for each problem. Make sure your name is onall three of your answer booklets. Please frame or circle your final answers to theproblems. Do not submit solutions to more than two problems. I f more than oneproblem is solved in an answer booklet ,only the first one will be accepted. You may notsubmit more than three answer booklets.

Electricity and MagnetismMultiple Choice Questions

Answer the following questions in a bluebook Write thenumber the question with your answer next to it

A parallel plate capacitor filled with a slab of linear homogeneous isotropic dielectric; So is first charged up by placing it across a battery Vo then disconnecting from thebattery. The dielectric is then removed from the capacitor. Which-ofthe followingstatements is false?

(a) The energy density of the ,electric. field decreases.- (b) The capacitance decreases by a factor K. (c) A force is required to remove the slab.(d) The voltage difference across the plates increases

2. Consider an inductance L a capacitor C, and a resistor R Which of the followingstcrtements is false?

(a) A charged capacitor across a resistor falls to 37 of its original voltage in a time, equal to RC(b) An circuit resonates at a frequency of(LC)-l.(c) The impedance of R in series powered by an AC source of angular frequency

cuis [ XL x i R2]1 2 where XL = cu and Xc = CUC -l.(d) When a voltage V is impressed across Rand L in series, voltage of magnitude V

is also found initially across the inductance opposing the flow o f current. -


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. , , ,shown below, entering and leaving through beam defining slits. Which statementbelow correctly describes all three devices.

;

IYI

1

oL E i points r ~ i l l yinward\ Capacitor plates .\ .

\

~ I. = = T ~

II . v E , are mutually perpendicular_ 1 _

I

I

v .l.. points iiito the pageMagnetic pole face

I- - - ~ o oJ

a) I velocity filter; II. energy filter; III. momentum filter.

b) I energy filter; II. momentum filter; III. velocity filter. c) momentum filter; II. velocity filter; III. energy filter. d) none of the above.

4 Consider two round flat metal plates, one with twice the area of the other. Each isplaced at a potential a relative to earth ground. The plates are then placed in contactwith their centers coincident on top of one another. The combined plates are now at apotential a) 2/3 b) a

c) 3/2 d) 2 a


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Thread

7 ;;; m/ s

Given a circular loop of copper wire suspended from a thread so it hangs freely, whatwould you observe as a permanent magnet is moved toward the center ofthe wire

loop at a constant velocity as depicted above?

a) The loop will swing away from the magnet. b) The loop will swing toward the magnet c) The loop will not move. d) There is insufficient information to predict the behavior of the loop.

6 To investigate the AC characteristics of a series RL circuit, a function generatorprovides a time-dependent source potential of V = sin 2nft and the potential drop

across an inductor is VL = L di/dt. The appropriate expression for the inductor sreactance, XL, and the RL circuit s impedance, Z is:

a Z = R + X L ; b) Z Sqrt R 2 Xl ; c) Z = Sqrt XLR ; d) Z = R XL cos 2rrft .

7 Capacitive pressure transducers are designed to sense the change of spacing betweena movable diaphragm and a fixed plate that form a capacitor. I f the pressure sensorhas two plates that have an area of 4x l 0 3 with an air gap separation o f d = 1 mmwhen the pressure is zero, what is the sensor s capacitance when, under an appliedpressure, the separation is reduced to 0.4 mm

a) 35 ~ Lb) 90 pF

c) 35 pF d) 90 ~


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having a radius ] is coaxial with a smaller coil of turns and radius R2 in the sameplane as shown. Choose the appropriate expression for the mutual inductance M forthis arrangement of these two coils., assuming R2 .

Taken from Fundamentals o f Physics by Haliday, Resnick Walker, John Wiley Sons, Inc.

Four point charges are located as follows: a charge q at x y = a 0), a charge q at -a, 0), a charge q at (0, a , and a charge -q at O,-a . Far away from these charges,what type of electric potential pattern will e dominant?

~ A monopole potential pattern will be dominant.(b) A dipole potential pattern will be dominant.(c) A quadrupole potential pattern will be dominant.(d) An octupole potential pattern will be dominant.

10. A circular ring of charge with positive linear charge density 1 and radius b iscentered on the origin and placed in the x-y plane. A solid, grounded, conductingsphere of radius a is also centered on the origin, where a b f we were to apply themethod o f images to solve for the electric potential everywhere external to the sphere,what image charge distribution must we use?

a Use a negative point charge at the origin. b Use a positive point charge at the origin. /(c) Use a negative circular ring of c h < r ~ einside the sphe e, centered at the origin.(d) Use a positive circular ring o f charge inside the sphere, centered at the origin.


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Vsin 3nx/a . Ifwe were to solve for the electric potential at any point inside th e box,how would this solution depend on x?

a) sin 3nx/ab) sin mnx/a for m = 4 5 6 c) cos 31LX/ad) cos mnx/a sin m1LX/a for m = 1 2 3 4 ...

2 A two-dimensional container is formed by th e intersection of circles of radii r = 2 and

r = 4 with th e positive x an d y axes, as shown in the figure. Th e container contains nocharge, bu t i ts walls are held at certain potentials. How many boundary conditions dowe need to un qu ly solve th e two-dimensional Laplace equation for th e electricpotential in the container, but not over-specify the problem?

a) 1 b 2 c) 3 d) 4

13. An e l e ~ t r o m g n e t iwave is propagating in free s pa ce . W hi ch one of the followingstatements about the wave is false?

a) The electric field is in phase with the magnetic field. b) The electric energy density equals th e magnetic energy density. c) The electric field is perpendicular to th e magnetic field. d) The wave propagates in the direction of the electric field.

4 An electromagnetic wave is propagating in a good conducting material. Which one ofth e fol lowing statements about the wave is false?

a) The wave is attenuated as i t propagates into the mater ial. b) The magnetic energy density is greater than the electric energy density. c) The electric field vector is in phase with the magnetic field vector. d) The wave propagates in a direction perpendicular to both the electric and

magnetic field vectors. ,


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. uygen s pr nc p e can e use o _

(a) derive the law of refraction but not the law of reflection;(b) derive the law of refraction and law of reflection;(c) show that light is a wave;(d) show that light is a particle.

19 A wave of light is incident on the interface between a solid dielectric material and air,and it experiences total internal reflection. f we could increase the dielectric constanto f the solid material until it became effectively infinite (but such that the wave couldstill propagate), how would this effect the total internal reflection phenomenon?

(a) The wave would be totally internally reflected at l l angles of incidence.(b) The wave would not e totally internally reflected at ny angles of incidence.(c) The wave would be internally reflected for incidence angles greater than 45.(d) The transmitted wave would flip to the other side of the normal.

2 Consider the following statement. f I take a green banana into a dark room andshine red light on it, it should appear yellow, that is ripe, because overlapping greenand red light produces yellow light. Which one of the following arguments eitherrefutes or supports this statement?

(a) The statement is correct because of the reason stated.(b) The statement is incorrect because green bananas not emit green light.(c) The statement is incorrect because overlapping green and red light do not produce

yellow light.(d) The statement is incorrect because green bananas will appear totally dark under

red light.


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Problems 1 and 2 Submit one bluebook for one problem.

Solve only one of the following two problems, either Problem 1 or Problem 2 but notboth.

Consider a spherical free charge distribution of radius R and total charge Q given by .the following charge density functions:

p = A for r < R/3

p = 3/2 A r/R for RI3 < r < R

p = 0 for R

where A Q and R are constants and r is the radial distance from the center of thecharge distribution.

a First determine the value of A in terms of Q and R.

b Obtain the electric field as a function of r in the two regions r < R/3 andR/3 < r< R

thick spherical shell oflinear isotropic homogeneous dielectric = K GO extendingfrom r = R to r = 3R surrounds this charge distribution. .

c What are the bound charge densities on the inner and outer spherical surfaces ofthe dielectric?

Calculate th potential difference between r = Rand r = 3R. I

2. Evidence of magnetic inductance can be observed simply by dropping a cylindricalrare earth magnet having a mass m and magnetic field B through a vertically orientedconducting tube and noticing the delayed manner in which the magnet falls. Themoving magnetic field of magnitude B gives rise to a circulating induced electriccurrent in the conducting tube of radius r while the circulating electric current givesrise to a magnetic field in opposition the moving magnetic field.

a Draw a figure and clearly define the physical parameters.b Write down the electromagnetic properties and constraint equations.c Derive the equation of motion as a function of resistance and the defined

parameters.d Derive an expression for the constant terminal velocity v at which the magnet will

fall as a function of the resistance R encountered by the circulating electriccurrent.


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Problems 3 and 4 Submit one bluebook for one problem.

Solve only one the following two problems .either Problem 3 r Problem 4 but notboth.

3. A point charge +q is placed near the interior comer o a semi-infinite groundedconducting object. The point charge is located at = a, y = a, and the conductingobject is defined by the positive and y axes as shown in the figure. Note that we are

only interested in the plane containing the point charge and nOlIDal to the surfaces sothat the problem is two-dimensional. For a point charge = q/ 47l:8or .

y

a Using the method o i m g ~ sdetermine the electric potential everywhere in thefirst quadrant in terms o Cartesian coordinates.

b What does this reduce down to we care only for points along the line = yVery near the comer a, what value does the potential approach?

\ r

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