Physics 8.02 Exam Two Equation Sheet Spring 2004

Physics 8.02 Exam Two Equation Sheet Spring 2004

surfaceclosed o

insideQ

dAE

outsidetoinsidefrompointsdA

b

a

abbtoafrommoving VVV dsE

0 dsE

N

i io

iqV

1 4 rrchargespointmany

V

QC

C

QV

U = 12

C V2 =

Q2

2C

I q n Awirevdrift

driftwire

Iq n

A J v

where is the resistivity E J

1/where the conductivity J E

V = i R

L LR

A A

1Rparallel

= 1R1

+ 1R2

Rseries = R1 + R2

R

VRiViP heatingohmic

22

(joules/sec)

2

ˆ

4o q

r

v rB

2

ˆ

4oI

r

ds rdB

ˆ points source observerr from to

througho

contour

I dsB

where Ithrough is the current flowing through the open surface bounded by the contour

center of circle of radius 2o

a

IB

a

surfaceopen

throughI dAJ

ds right-handed wrt dA F q v Bext

extI dF ds B

Cross-products of unit vectors:

ˆ ˆ ˆ ˆ ˆ ˆ 0 i i j j k k

ˆ ˆ ˆ i j k ˆ ˆˆ i k j ˆ ˆˆ j k i

Useful integrals: 1/ 2

1/ 2

2( )

( )

dc f

c f f

1ln( )

( )

dc f

c f f

2

1 1

( ) ( )

d

c f f c f

Useful Small Argument Approximations (1 ) 1 for <<1n n ln(1 ) for <<1

0vol

all space

U dV2

E E

8.02 2nd Exam Name_____________________

Problem 1 (25 Points) Circle your choice for the correct answer to the eight questions below.

A: A current I is uniformly distributed over the cross-section of a conducting wire of radius a. For distances r from the center of the wire with r > a,

1. The magnitude of the magnetic field does not depend on r. 2. The magnitude of the magnetic field is proportional to 1/r 3. The magnitude of the magnetic field is proportional to r 4. The magnitude of the magnetic field is proportional to 1/r2 5. The magnitude of the magnetic field is proportional to r2

B: A segment of a wire of length dl carries a current I (see sketch below).

The magnitude of the magnetic field due to this wire segment at the location P shown in the sketch is

1. 2/3224 ba

dlaIo

2. 224 ba

dlaIo

3. 2/3224 ba

dlbIo

4. 224 ba

dlbIo

5. 224 ba

dlIo

8.02 2nd Exam Name_____________________

A: Eight parallel wires cut the page perpendicularly at the points shown. All the wires carry the same magnitude of current 0I , but some carry current into the page and some

carry current out of the page. For wires 1 to 4, the current flows up out of the page; for

the rest, the current flows down into the page. If we valuate d B s along the closed

path (see figure) in the direction indicated by the arrowhead, we get

(a) 4 o oI (b) 3 o oI (c) 2 o oI (d) o oI (e) 0

(f) o oI (g) 2 o oI (h) 3 o oI (i) 4 o oI

F: A wire loop carries current I1, and is located near an infinite wire carrying current I2. The currents flow in the directions shown. The net force on the wire loop due to the presence of the infinite wire is

(a) upwards (b) downwards (c) 0 (d) to the left (e) to the right

G: The coil below has current flowing clockwise as seen from above. It sits in an external magnetic field shown by the field lines below. The coil will feel a magnetic force that is

8.02 2nd Exam Name_____________________

(a) Upwards (b) Downwards (c) Zero

8.02 2nd Exam Name_____________________

Problem 2 (25 points): A Bainbridge mass spectrometer is shown in the figure. A charged particle with mass m, charge q and speed V enters from the bottom of the figure and traces out the trajectory shown in the fields shown. The only electric field is in the region where the trajectory of the charge is a straight line. (a) Is the sign of the charge positive or negative? To get credit for your answer you must give a logical reason to justify it.

(b) When the particle is moving through the first (straight-line) segment of its trajectory, what is its speed V, in terms of E and B? Justify your answer.

8.02 2nd Exam Name_____________________

(c) When the charge is moving through the second (curved) segment of its trajectory, derive a relation between the radius r of the arc through which it moves, its charge q, its speed V, and Bo. To get credit for your answer, you must show the details of your derivation. (d) The charge hits the left wall of the spectrometer at a vertical distance h above where it entered the upper region and a horizontal distance L to the left of where it entered the upper region (see sketch). Given this information and your result from above, what is the speed of the particle in terms of h, L, q, m, and Bo?

8.02 2nd Exam Name_____________________

Problem 3 (25 points): A wire segment is bent into the shape of an Archimedes spiral (see sketch). The equation that describes the curve in the range 0 (see sketch) is

( ) , for 0b

r a

where is the angle from the x-axis in radians ( radians is 180 degrees). The point P is located at the origin of our xy coordinate system. The vectors ˆ re and ˆe are the unit

vectors in the radial and azimuthal directions, respectively, as shown. The wire segment carries current I, flowing in the sense indicated.

(a) A infinitesimal line segment ds

of the spiral is located at a distance r from the

origin and at an angle from the x-axis. What is the magnetic field dB

at the point P due to this current carrying segment of the wire? You may use the expression

ˆ ˆrdr r d ds e e

. Give both the magnitude and direction.

8.02 2nd Exam Name_____________________

(b) What is the total magnetic field B

at point P due to the entire Archimedes spiral

segment between 0 ? You will find useful one of the integrals on the formula sheet.

(c) Suppose the distance b goes to zero. What does your expression above become in

this limit? Is this what you expect? Justify your answer. You will find useful one of the small argument approximations given on the formula sheet.

8.02 2nd Exam Name_____________________

Problem 4 (25 points): A coaxial cable consists of a solid inner conductor of radius a, surrounded by a concentric cylindrical tube of inner radius b and outer radius c. The conductors carry equal and opposite currents I0 distributed uniformly across their cross-sections. Determine the magnetic field at a distance r from the axis for the following ranges of radii. On the figure below, draw the amperean loop you use in each case.

(a) r < a (b) a < r < b

8.02 2nd Exam Name_____________________

(c) b < r < c (d) Plot your answers for the magnitude of B above on the graph below. Label your vertical axis or you will lose points.

8.02 2nd Exam Name_____________________

Problem 2 (25 points): A current I flows around a continuous path that consists of portions of two concentric circles of radii R and R/2, respectively, and two straight radial segments. The point P is at the common center of the two circle segments. (a) Use the Biot-Savart Law, to calculate dB at P due only to that segment of the path ds shown in the sketch. Indicate on the sketch the vector r̂ you use and the direction of dB. Give the magnitude of dB in terms of I, R, dl, and o.

(b) Using your expression in (a), find the magnetic field at P due to the larger circle segment only. Give its magnitude and direction. (c) What is the total field B at P? Give its magnitude and direction.

8.02 2nd Exam Name_____________________

Problem 4 (25 Points) Circle your choice for the correct answer to the five questions below. A: A particle with known charge q and known mass m is performing circular motion in a uniform magnetic field of known magnitude B (as usual, the orbit is perpendicular to the magnetic field).

With q, m, and B known, examine the following statements: (a) The radius R of the orbit can be determined uniquely. (b) The speed v of the particle can be determined uniquely. (c) The angular velocity ω of the particle can be determined uniquely.

Circle the one correct statement below

1. Only (a) is correct. 2. Only (b) is correct.

3. Only (c) is correct

4. Only (a) and (b) are correct.

5. Only (a) and (c) are correct.

6. Only (b) and (c) are correct.

7. All are correct.

8. None is correct

B: Consider an infinitely long cylindrical solenoid of radius R with n turns per unit length and a current I. The magnetic field due to this solenoid satisfies:

1. 0B for r < R and 0B nI for r > R.

2. 0B nI for r < R and 0B for r > R.

3. 0

1

2B nI for r < R and 0B for r > R.

4. 0B for r < R and 0

1

2B nI for r > R.

5. 0B nI for all r.

8.02 2nd Exam Name_____________________

C: Consider a charged circular loop of radius R and linear charge density λ (charge per unit length) glued down to the loop. Suppose the loop is rotating with angular velocity ω about the axis normal to the loop and going through its center. The loop acts as a magnetic dipole with magnetic dipole moment given by

1. 2R

2. 2R

3. 22 R

4. 3R

5. 32 R

D: An infinitely long wire carries a current I. What is the magnitude B of the magnetic field a distance r away from the wire?

1. 0

2

IB

r

2. 0IB

r

3. 0IB

r

4. None of the above. A: A charged particle of mass m and charge q has a speed v and moves in a circular orbit in a magnetic field of strength B.

1. a noticeable force and no noticeable torque 2. no noticeable force and a noticeable torque 3. a noticeable force and a noticeable torque 4. no noticeable force and no noticeable torque

B: A current I is uniformly distributed over the cross-section of a conducting wire of radius a. For distances r from the center of the wire with r < a,

6. The magnitude of the magnetic field is constant. 7. The magnitude of the magnetic field varies as 1/r 8. The magnitude of the magnetic field varies as r 9. The magnitude of the magnetic field varies as 1/r2 10. The magnitude of the magnetic field varies as r2

8.02 2nd Exam Name_____________________

C: A segment of a wire of length L carries a current I (see sketch above). The magnitude of the magnetic field a distance z from the center of the wire segment along its perpendicular bisector (see sketch) is given by the expression

6.

2/

2/ 2/3224

L

L

o

zx

dxxI

7.

2/

2/ 224

L

L

o

zx

dxxI

8.

2/

2/ 2/3224

L

L

o

zx

dxzI

9.

2/

2/ 224

L

L

o

zx

dxzI

10.

2/

2/ 224

L

L

o

zx

dxI

I. The sketch shows a current carrying element of wire. The element has length ds, cross-sectional area A, and carries current I. It sits in a constant external magnetic field Bext. The charge carriers in the wire have number density per unit volume n, positive charge q > 0, and drift speed vdrift. Starting from the Lorentz force law for the force on a single charge carrier, F q vdrift Bext , derive an expression for the total magnetic force

on this current carrying line element in terms of I, ds, and Bext. You must show the steps you use in arriving at this formula to be given credit.

8.02 2nd Exam Name_____________________

II. The sketch shows a resistor with resistance R, length L and cross-sectional area A. It is made out of material with resistivity . Starting from Ohm’s Law in microscopic form, E = J, derive the relationship between R, L, A, and You must show the steps you use in arriving at this formula to be given credit.

III. The sketch shows two co-axial current-carrying rings. The rings have current flowing in them in the same sense, as shown (clockwise as viewed from the top). Are these rings attracted to each other, repelled by each other, or do they feel no mutual force? You must explain how you arrived at this answer to be given credit.

8.02 2nd Exam Name_____________________

E: Two wires run parallel to the z-azis, which is out of the page. The wire on the right carries a current I1 > 0 out of the page. The wire on the left carries a current of I2 = I1 /2 also out of the page. Which of the four “iron filings” representation of the magnetic field of these two wires shown below is correct? In an “iron filings” representation, the magnetic fields are parallel (or anti-parallel) to the “streaks”.

(a) (b)

(c) (d)

8.02 2nd Exam Name_____________________

Problem 3 (25 points): For a finite wire carrying a current I the contribution to the magnetic field at point P is

01 2( ) (cos cos )B r

r

where B is the magnitude of the magnetic field and 1 and 2 are the angles indicated in

the figure:

Consider a current I on a piece of wire of length 2a: (see figure below).

(a) Find the magnitude B of the magnetic field at point P located a distance r away from the wire and equidistant from the endpoints of the wire. On the above diagram indicate with an or whether the field points in or out of the page.

8.02 2nd Exam Name_____________________

Consider a square loop of side 2a lying on the (x, y) plane as illustrated on the figure below (corners at (x, y) = ( , )a a and ( , )a a ). We are interested in the magnetic field at the point P on the z axis, a distance z from the origin.

(b) What is the magnitude 1B of the magnetic field at P contributed separately by each

of the sides of the loop. Give your answer in terms of a, z, and other constants ( 0 , I, ,

...) [You may wish to use the result of (a).]

8.02 2nd Exam Name_____________________

(c) Find the magnitude tB of the total magnetic field at P. What direction does tB

point? (d) Calculate the leading value of the magnetic field tB far away from the loop, namely,

when z >> a. Write this magnetic field in terms of z and the dipole moment m of the current loop (as well as constants 0 , ,…).

8.02 2nd Exam Name_____________________

Problem 4 (25 points): A long cylindrical cable consists of a conducting cylindrical

shell of inner radius a and outer radius b. The current density J

in the shell is out of the

page (see sketch) and varies with radius as ( ) o

RJ r J

r for a r b and is zero outside

of that range. Find the magnetic field in each of the following regions, indicating both magnitude and direction. Show your work and your Amperean loops.

(a) r < a (b) a < r < b

8.02 2nd Exam Name_____________________

(c) r > b . Give also ( )B r b , the value of the magnetic field at r b . (d) Plot your previous answers for the magnitude of B on the graph below.

8.02 2nd Exam Name_____________________

Problem 1: Three Short Questions (25 points total):

8.02 2nd Exam Name_____________________

Problem 2: Biot-Savart Law and Magnetic Dipole Moment (25 points)

A current I flows around a continuous path that consists of portions of two concentric circles of radii R and 3R, respectively, and two straight radial segments, as shown. The point P is at the common center of the two circle segments. (a) What is the magnitude and direction of the magnetic field B at P due to the outer circle segment? Use the Biot-Savart Law to calculate this field. Show your work.

(b) What is the magnitude and direction of the magnetic field B at P due to the two straight line segments of the current loop? Use the Biot-Savart Law to calculate this field. Show your work. (c) What is the magnitude and direction of the magnetic field B at P due to the inner circle segment? (d) What is the total field B at the point P due to the entire current loop. Be sure to specify the direction of B. (e) What is the magnetic dipole moment of this current loop? Give its magnitude and direction.

Physics 8.02 Exam Two Equation Sheet Spring 2004

Documents

Transcript of Physics 8.02 Exam Two Equation Sheet Spring 2004