Electromagnetism. KU. Concept Test Practice

8/3/2019 Electromagnetism. KU. Concept Test Practice

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1

Answers are at the end

What is (from r 1 to r)?

q1

r 1=(x1,y1)

r = (x,y)

1ℜ̂

1 1ℜ = −r rv

ˆ A =r

A / | A |

1

( ) ( ) ( ) ( )

1 1 1 1

1 1 1 1

2 2 2 21 1 1 1

A) (x x , y y ) B) (x x, y y)

(x x , y y ) (x x, y y)C) C)

x x y y x x y y

E) None of these

− − − −

− − − −

− + − − + −

Coulomb's law :

In the fig, q 1 and q 2 are 2 m apart.Which arrow can represent ?

1 2122

12

ˆ(by 1 on 2) = ℜℜ

Fr kq q

ℜ̂12

q1 q2

A

B C

D) More than one (or NONE) of the aboveE) You can't decide until you know if q 1 and q 2

are the same or opposite signed charges

2

Can I always use the Coulomb law in thisform to calculate the force on a small chargeat any point in vacuum if I know the locationof all charges for all times? (Assume noconductors or dielectrics are present.)

A) Yes, of course! It’s a law and laws

are always true.B) No. The coulomb law works only forspecific situations.

C) I don’t know and my neighbor hasno clue either.

3

Two charges +q and -q are on the y-axis,symmetric about the origin. Point A is an emp typoint in space on the x-axis. The direction of theE field at A is…

A.UpB.Down

C.LeftD.RightE.Some other direction, or E = 0, or ambiguous

4

+q

x

y

-q

A

5 charges, q, are arranged in aregular pentagon, as shown.What is the E field at the center?

A) ZeroB) Non-zeroC) Really need trig and a calculator to

decide

q

q

qq

q

5

1 of the 5 charges has been removed, asshown. What’s the E field at the center?

q

qq

q

A) +(k ·q/a 2) jB) -(k ·q/a 2) jC) +(4 ·k·q/a 2)jD) -(4 ·k·q/a 2)jE) Something entirely different!

+x

+ya

6

To find the E- field at P from a thin line

(uniform linear charge density λ):

A) x B) y'

C) D)

E) Something completely different!!

E = 14πε 0

1

ℜ

2∫ ℜ̂ λ dl'

P=(x,0,0)x

y

dl'

22 ' y x +22' xdl +

r'

r

7

ℜ=ℜv

What is ?

,so

P=(x,0,0)

x

y

dl'

r

ℜr ' = (0,y',0)

r

r

E (

r

r ) =λ dl'

4πε 0ℜ3∫

r

ℜ ∫ =K

04)0,0,( πε λ

x E x

elseSomething))'(''

)

'')

)'('

)

')

2 / 322

3

2 / 322

3

E y x

ydy D

x ydy

C

y x xdy

B

x xdy

A

∫

∫

∫

∫

+

+

8



2

D) None of these

To find E at P from a negatively charged sphere(radius R, uniform volume charge density ρ) using

what is (given the smallvolume element shown)?

E =1

4πε 0

1

ℜ

2∫ ˆℜ ρ dτ '

P=(X,Y,Z)

x

yz(x,y,z)

R A

BC

r

ℜ

9

A)

B)

C)

D) E) None of these

E = 14πε 0

1

ℜ

2∫ ℜ̂ ρ dτ P=(X,Y,Z)

X ,Y , Z ( )( X − x)

2

+ (Y − y)2

+ ( Z − z)2

∫

ρ dxdydz

X ,Y , Z ( )( X − x)2 + (Y − y) 2 + ( Z − z)2( )3/2∫ ρ dxdydz

X − x,Y − y, Z − z( )( X − x)

2

+ (Y − y)2

+ ( Z − z)2

∫

ρ dxdydz

X − x,Y − y, Z − z( )( X − x)2 + (Y − y)2 + ( Z − z)2( )3/2

∫ ρ dxdydz

= 14πε 0

• (....?)

x

yz(x,y,z)

R

10

A positive point charge +q is placed outside a closedcylindrical surface as shown. The closed surfaceconsists of the flat end caps (labeled A and B) and thecurved side surface (C). What is the sign of the electricflux through surface C?

(A) positive (B) negative (C) zero(D) not enough information given to decide

(Side View)

q

q

A

B

C

11

A) σ/2ε 0B) σ/ε 0C) 2 σ/ε 0D) 4 σ/ε 0E) It depends on the

choice of surface

12

Given a pair of very large, flat, chargedplates with surface charge densities + σ.Using the two Gaussian surfaces shown(A and B), what is the E field in the regionOUTSIDE the plates?

+ + + + + + + + + + + + + + + +

+σ

+σ

A B

+ + + + + + + + + + +

+ + + + +Do it on your white board!

You have an E field given byE = c r / ε0, (Here c = constant,

r = spherical radius vector)

What is the charge density ρ(r)?

A) c B) c r C) 3 c D) 3 c r^2E) None of these is correct

13

Given E = c r / ε0,(c = constant, r = spherical radius vector)On Wednesday we found ρ(r) = 3c.What is the total charge Q enclosed by animaginary sphere centered on the origin,of radius R?

Hint: Can you find it two DIFFERENT ways?

A) (4/3) π c B) 4 π cC) (4/3) π c R^3 D) 4 π c R^3E) None of these is correct

2.2114

What are the units of δ(x) if x ismeasured in meters?

A) δδδδ is dimension less (‘no units’)B) [m]: Unit of lengthC) [m 2]: Unit of length squaredD) [m -1]: 1 / (unit of length)E) [m -2]: 1 / (unit of length squared)

15

What are the units of δ3(r ) if thecomponents of r are measured in

meters?

A) [m]: Unit of lengthB) [m2]: Unit of length squaredC) [m -1]: 1 / (unit of length)D) [m -2]: 1 / (unit of length squared)E) None of these.

16

A point charge q is at pos ition R, as shown.What is ρ(r), the charge density in all space?

q

origin

R3

3

3

3

A) ( ) q ( )

B) ( ) q ( )

C) ( ) q ( )

D) ( ) q ( )

E) None of these or More than one of these

ρ = δ

ρ = δ

ρ = δ −

ρ = δ −

r R

r r

r r R

r R r

r r

r r

r r r

r r r

17



3

An infinite rod has uniform chargedensity λ. What is the direction of the Efield at the point P shown?

Origin

P

A) 45 °

B) Less than 45 °C) horizontalλ

18

A point charge q is at position R, as shown.What is ρ(r), the charge density in all space?

q

origin

R3

3

3

3

A) ( ) q ( )

B) ( ) q ( )

C) ( ) q ( )

D) ( ) q ( )

E) None of these or More than one of these

ρ = δ

ρ = δ

ρ = δ −

ρ = δ −

r R

r r

r r R

r R r

r r

r r

r r r

r r r

19

M

Consider the 3D vector field

in spherical coordinates,where c = constant .

2

ˆV(r) c

=

r r

The divergence of this vector field is

:

A) Zero everywhere except at the originB) Zero everywhere including the originC) Non-zero everywhere, including the origin.D) Non-zero everywhere, except at origin (zero at origin)

(No fair computing the answer. Get answer from your brain.)

20

Consider the vector field

where c = constant .

ˆV(r) c r=r

r

R I,( )The divergence of this vector field

is:A) Zero everywhere except at the originB) Zero everywhere including the originC) Non-zero everywhere, including the origin.D) Non-zero everywhere, except at origin (zero at origin)

(No fair computing the answer. Get answer from your brain.)

21

The voltage (or ‘potential V(r)’) is ze ro at a point inspace.

You can conclude that :A) The E-field is zero at that point.B) The E-field is non-zero at that pointC) You can conclude nothing at all about the E-

field at that point

22

We usually choose V(r ∞ ) ≡ 0 whencalculating the potential of a point charge to beV(r) = + κ q/r. How does the potential V(r)change if we choose our reference point to beV(R)=0 where R is closer to +q than r.

A V(r) is positive but smaller than kq/rB V(r) is positive but larger than kq/rC V(r) is negativeD V(r) doesn’t change (V is independent of

choice of reference)

+q R r ∞

23

A uniformly charged ring, in the xy plane, centered on the

origin, has radius a and total charge Q = 2 π·λ· aV(r = ∞) = 0. What is the v oltage at z on the z-axis?

a

z 2 22 2

k Q k QA) B)

a zk Q k Q

C) D)a za z

E) None of these

++

24

Given a sphere with uniform surface

charge density σ (and no other chargesanywhere else) what can you say aboutthe potential V inside this sphere?(Assume as usual, V( ∞ )=0)

A) V=0 everywhere insideB) V = non-zero constant everywhere

insideC) V must vary with position, but is zero

at the center.D) None of these.

25

A) Could be E(r), or V(r)

B) Could be E(r), but can't be V(r)

C) Can't be E(r), could be V(r)

D) Can't be either E) ???

Could this be a plot of |E|(r)? Or V(r)? (forSOME physical situation?)

26







6

Two very strong (big C) idealcapacitors are well separated.

-Q Q -Q Q

++++++++++++

------------

++++++++++++

------------

A)YesB)NoC)???

45

If they are connected by 2 thin conductingwires, as shown, is this electrostaticsituation physically stable?

A) 0 B)

C)

D) Something more complicated

A point charge +Q sits above a very large grounded conducting slab .What's the electric force on +Q?

+Q

Q 2

4πε 0d 2downwards

Q2

4πε 0 (2 d )2 downwards

46

z

xy

d

Which of the following cases could actually occurabove and below a sheet of surface charge?

47

+++++++++++++++++

+++++++++++++++++

+++++++++++++++++

+++++++++++++++++

E1

E2

E1E1

E1

E2 E2

E2

A

E: None of these

C D

B

A)

B) Something else.

A point charge +Q sits above a very large grounded conducting slab .What's the energy of this system?

+Q

48

z

xy

d

−Q 2

4πε 0(2 d )

Sinh and cosh

Two solutions for positive C are sinh x and cosh x :

- 2 - 1 1 2

- 3

- 2

- 1

1

2

3

1

2

3

-3

x

Which is which?A)Curve 1 is sinh x and curve 2 is cosh xB)Curve 1 is cosh x and curve 2 is sinh x

49

Is this a stable charge distribution fortwo neutral, conducting spheres?

++++

+

-----

++++

+

-----

A) YesB) No C) ???

50

A) Simple Coulomb’s law:

B) Something more complicated

A point charge +Q sits above a very

large grounded conducting slab .What is E(r) for points above the slab?

zxy

+Qd

r

E (r

r ) =Q

4πε 0

r

ℜ

ℜ3 with

r

ℜ = (r

r − d ˆ z)

51

r

Say you have three functions f(x), g(y) and h(z).

f(x) only depends on ‘x’ but not on ‘y’ and ‘z’.g(y) only depends on ‘y’ but not on ‘x’ and ‘z’.h(z) only depends on ‘z’ but not on ‘z’ and ‘y’.

If f(x) + g(y) + h(z) = 0 for all x, y, z, then:A) All three functions are constants (i.e. they do

not depend on x, y, z at all.)B) At least one of these functions has to be zero

everywhere.C) All of these functions have to be zero

everywhere.D) All three functions have to be linear functions in

x, y, or z respectively (such as f(x)=ax, a ≠0 etc.)

52

where C 1+C 2 = 0. Given the boundaryconditions in the figure, which coordinateshould be assigned to the negative constant(and thus the sinusoidal solutions)?

1Y

d 2

Y dy 2 = C 2 1

X d

2

X dx 2 = C 1

A) x B) y

C) C1= C 2=0 here

D) It doesn’t matter

53Given the two diff. eq's:



7

where C 1+C 2 = 0. Which coordinate shouldbe assigned to the negative constant (andthus the sinusoidal solutions)?

1Y

d 2Y dy 2 = C 2

1 X

d 2 X dx 2 = C 1

A) x B) y

C) C1= C 2=0 here



where C 1+C 2 = 0. Which coordinate shouldbe assigned to the negative constant (andthus the sinusoidal solutions)?

1Y

d 2Y dy 2 = C 2

1 X

d 2 X dx 2 = C 1

A) x B) y

C) C1= C 2=0 here



56 The X(x) equation in this problem involvesthe "positive constant" solutions:A sinh(kx) + B cosh(kx)

What do the boundary conditions sayabout A and B?A) A=0 (pure cosh)

B) B=0 (pure sinh)

C) Neither: you shouldrewrite this in terms ofA ekx + B e -kx !

D) Other/not sure?

What is the value of ?

A) ZeroB) πC) 2 πD) π/2E) Something else/how could I

possibly know this?

∫ π 2

0

)3sin()2sin( dx x x

57Suppose V 1(r) and V 2(r) are linearlyindependent functions which both solveLaplace's equation,Does aV 1(r)+bV 2(r) also solve it (with ‘a’and ‘b' constants)?

02 =∇ V

A) Yes. The Laplacean is a linear operatorB) No. The uniqueness theorem says this

scenario is impossible, there are nevertwo independent solutions!

C) It is a definite yes or no, but thereasons given above just aren't right!

D) It depends...

58

How does V(x,y) compare, 4 m above the middleof the base in the two troughs?A) Same in eachB) 4x bigger in #1 C) 4x bigger in #2D) much bigger in #1 E) much bigger in #2

59

x

y

V ( x, y) =

4V 0π

1nn =1,3,5...

∞

∑ sin( nπ x / a )e−nπ y / a

Can you write the functionas a sum of Legendre Polynomials?

P0(cos θ ) =1, P

1(cos θ ) = cos θ

P2 (cos θ ) = 32

cos 2 θ − 12

, P3 (cos θ ) = 52

cos 3 θ − 32

cos θ

)cos1( 20 θ +⋅V

A)No, it cannot be doneB) It would require an infinite sum of termsC) It would only involve P 2D) It would involve all three of P 0, P 1 AND P 2E) Something else/none of the above

60

V 0 (1+ cos 2 θ ) =???

C l Pll= 0

∞

∑ (cos θ )

Suppose V on a spherical shell isconstant, i.e. V(R, θ) = V0.Which terms do you expect to appearwhen finding V(outside) ?A) Many A l terms (but no B l's)B) Many B l terms (but no A l's)C) Just A 0D) Just B 0E) Something else!!

)(cos),(0 1

θ θ

∑

∞

= +

+=

l ll

ll

lP

r

Br Ar V

61

Suppose V on a spherical shell isconstant, i.e. V(R, θ) = V 0.Which terms do you expect to appearwhen finding V(inside) ?A) Many A l terms (but no B l's)B) Many B l terms (but no A l's)C) Just A 0D) Just B 0E) Something else!

)(cos),(0 1

θ θ

∑

∞

= +

+=

l ll

ll

lP

r

Br Ar V

62



8

Suppose V on a spherical shell is

Which terms do you expect to appearwhen finding V(inside) ?A) Many A l terms (but no B l's)B) Many B l terms (but no A l's)C) Just A 0 and A 2D) Just B 0 and B 2E) Something else!

V (r ,θ ) = Al r l +

Bl

r l +1

Pll= 0

∞

∑ (cos θ )

V ( R,θ ) = V 0 (1 + cos 2 θ )

63

Boundary conditions: sigma

How many boundary conditions (onthe potential V) do you use to find V

inside the thin plastic sphericalshell?

A) 1B) 2C) 3D) 4E) depends on σ 0

64

σ 0 (θ )

Boundary conditions: sigmaHow many boundary conditions (onthe potential V) do you use to find Vfor this thin plastic spherical shell?

A) 1B) 2C) 3D) 4E) depends on σ 0

65

σ 0 (θ )

For a dipole at the origin pointing in the z-direction, wehave derived

( )dip 30

p ˆˆ(r ) 2cos sin4 r

= θ + θ θπε

E rr

v

For the dipole p = q d shown, what does theformula predict for the direction of E(r=0)?

66

x

z+

-

d

A)Down B) Up C) some other direct ion

D) The formula doesn't apply.

At the end of last class we deriv ed the potential fora dipole at the origin pointing in the z-direction.Using E = -∇∇∇∇V we can find the E-field in sp hericalcoordinates:

( )dip 30

p ˆˆ(r) 2cos sin4 r

= θ + θ θπε

E rr

v

For the dipole p = q d shown, whatdoes the formula predict for thedirection of E(r=0)?

67

x

z+

-

d

A)Down B) Up C) some other direc tion

D) The formula does n't apply.

For a collection of point charges , the dipolemoment is defined as

Consider the two charges, +2q and –q, shown.Which statement is true?

i ii

p q r= ∑v v

d

+2q

r2

r1

-q

x

yA) The dipole moment is

independent of the origin.B) The dipole moment

depends on the position ofthe origin.

C) The dipole moment is zero.D) The dipole moment is

undefined.

68

Griffiths argues that the force on a neutraldipole in an external E field is:

ext EpFr r

r r

)( ∇•=

If the dipole p points in the z direction, whatdirection is the force?

A) Also in the z directionB) perpendicular to zC) it could point in any directionD) the force is zero because the dipole is neutral

69

Griffiths argues that the force on a neutraldipole in an E field is: ext EpFr r

r r

)( ∇•=

If the dipole p points in the z direction, whatcan you say about E if I tell you the force isin the x direction?

A) E simply points in the x directionB) Ez must depend on xC) Ez must depend on zD) Ex must depend on xE) Ex must depend on z

70

You have a physical dipole, +q and -q afinite distance d apart.When can you use the expression:

A) This is an exact expression everywhere.

B) It's valid for large r

C) It's valid for small r

D) ?

V (v

r ) =1

4πε 0

v

p ⋅ r̂ r 2

71



9

Which charge distributions below producea potential which looks like C/r 2 (=dominantterm in V( r)) when you are far away?

E) None of these, or more than one of these!

(Note: for any which you did not select, howDO they behave at large r?)

72

Which charge distributions below producea potential which looks primarily like C/r 2

when you are far away?

E) None of these, or more than one of these!

(Note: for any which you did not select, howDO they behave at large r?)

73

The cube below (side a) has uniformpolarization P 0

(which points in the z direction.)What is the total dipole moment of this cube?

z

x

A) zeroB) a 3 P 0C) P 0D) P 0 /a 3

E) 2 P 0 a 2

74

The sphere below (radius a) has uni formpolarization P 0 (which points in the zdirection.)What is the total dipole moment of thissphere?

A) zeroB) P 0 a 3

C) 4 πa 3 P 0 /3D) P 0E) None of these/must be more

complicated

75

P 0

Are σb and ρb due to real charges?

A) Of course not! They are as fictitious as itgets! (Like in the ‘method of images.’)

B) Of course they are! They are as real as

it gets! (Like σ and ρ in Chapter 2.)C) I have no idea L

76

A dielectric slab (top area A, height h) hasbeen polarized, with P=P 0 (in the +z direction)What is the surface charge density, σb, on thebottom surface?

A) 0B) -P 0

C) P 0

D) P 0 A hE) P 0 A

77

P 0

In the following case, is the bound surfaceand volume charge zero or nonzero?

Physical dipoles idealized dipolesA) σb = 0, ρb≠0B) σb ≠ 0, ρb≠0C) σb = 0, ρb=0D) σb ≠ 0, ρb=0

78

In the following case, is the bound surfaceand volume charge zero or nonzero?

Physical dipoles idealized dipoles

79

A) σb = 0, ρb≠0B) σb ≠ 0, ρb≠0C) σb = 0, ρb=0D) σb ≠ 0, ρb=0

A stationary point charge +Q is near a block ofpolarizable material (a linear dielectric). The netelectrostatic force on the block due to the pointcharge is

++Q

A) attractive (to the left)

B) repulsive (to the right)

C) zero

80







12

A) B)

C) D)

E) None of the above

K = I / a 2 K = I / a

K = I /(4 a ) K = I /(a 2 L)

Current I flows down a wire (length L)with a square cross section (side a)

If it is uniformly distributed over theouter surfaces only, what is themagnitude of the surface currentdensity K ?

99

A "ribbon" (width ‘a’) of surfacecurrent flows (with surface currentdensity ‘K’)Right next to it is a second identicalribbon of current.Viewed collectively, what is thenew total surface current density?

A) KB) 2KC) K/2D) Something else

a

100A "ribbon" (width a) with uniform surfacecurrent density K passes through a uniformmagnetic field B ext. Only the length b alongthe ribbon is in the field. What is the

magnitude of the force on the ribbon?

A)KB

B) aKB

C) abKB

D) bKB/a

E) KB/(ab)

101

a

b

Bext

K

Is the total net charge in the universeconserved? How about the total mass?

A) Charge is conserved; total mass is conservedB) Charge is conserved; total mass is not conservedC) Charge is not conserved; total mass is conservedD) Charge is not conserved; total mass is not conservedE) Dude! How should I know?

102

Charge Conservation

Which of the following is a statementof charge conservation?

A) B)

C) D)

E) Not sure/can't remember

∂ ρ ∂t

= −r

J • d r

A∫∫

∂ ρ ∂t

= − (∇ •r

J )∫∫∫ d τ

∂ ρ ∂t

= −∇ •r

J

103

∂ρ ∂ t

= −r

J • d r

l∫ In the figure, with “dl” shown, which purplevector best represents ?

To find the magnetic field B at P due to acurrent-carrying wire we use the Biot-Savart law, v

B(v

r ) =µ 04π

I d

v

l × R̂R 2∫

ℜ

104

E) None of these is close!

What is the magnitude of

?

To find the magnetic field B due to acurrent-carrying wire, below, we use theBiot-Savart law,

a) b)

c) d) e)

dl sin θ ℜ

2

dl sinθ ℜ

3

dl cos θ ℜ

2dl cosθ ℜ

3

dlℜ

2

2

ˆ

R

R×ld v

∫ ×= 20 ˆ

4)(

R

Rld I r B

v

v v

π µ

ℜ

(And, what's here?)ℜ

105

What is the value of

?

To find the magnetic field B due to acurrent-carrying wire, below, we use theBiot-Savart law,

a) b)

c) d) e)

2

ˆ

R

R×dl I r

∫ ×= 20 ˆ)(

R

Rdl I r B

r

v v

π µ

ℜ

106

Iydx ' ˆ z[( x') 2 + y 2 ]3/ 2

− Iydx ' ˆ z[( x ') 2 + y2 ]3/ 2

− Ix ' dx ' ˆ y[( x') 2 + y 2 ]3/ 2

Ix 'dx ' ˆ y[( x ') 2 + y2 ]3/ 2

− Idx '( yˆ y − x' ˆ x)[( x') 2 + y 2 ]3/ 2

What is the direction of the infinitesimalcontribution dB(P) created by current in dl?

To find the magnetic field B at P due to acurrent-carrying wire we use the Biot-Savartlaw,

A) Up the pageB) Directly away from d l

(in the plane of the page)C) Into the pageD) Out of the pageE) Some other direction

v

B(v

r ) =µ 04π

I d

v

l × R̂R 2∫

107



13

How about direction of dB(P) generated JUSTby the segment of current dl in red?

A) B(p) in plane of page, ditto for dB(P, by red)B) B(p) into page, dB(P, by red) into pageC) B(p) into page, dB(P, by red) out of pageD) B(p) complicated - has mult component ( not ⊥or ||

to page), ditto for dB(P, by red)E) Something else!!

108What do you expect for direction of B(P)?

s

What is B at the point shown?

I

I

A)

B)

C)

D)

E) None of these

µ 0

π s I

µ 04π s

I

µ 02π s

I

µ 08π s

I (What directiondoes it point?)

109

I1 I2

I have two very long, parallel wires each carrying acurrent I 1 and I 2, respectively. In which direction isthe force on the wire with the cu rrent I 2?

(How would your answer change if you would reverse thedirection of the currents?)

A)UpB) DownC) RightD) LeftE) Into or out of the page

110

What is the magnitude of ?

To find the magnetic field B due to a current-carryingloop, we use the Biot-Savart law,

A) B)

C) D)

E) Something quite different!

dl sin φ z2

d v

l × R̂R 2

v

B(v

r ) = µ 0π

I d

v

l × R̂R 2∫

111

dl sin φ ( z2 + a 2 )

dl( z2 + a 2 )

dl z2

(Which colored arrow is ? r? r ’? )ℜ

To find the magnetic field B due to acurrent-carrying loop, we use the Biot-Savart law,

What is the dB z (the contribution to the verticalcomponent of B from this dl segment?)

A) B)

C) D)

E) Something quite different!

v

B(v

r ) = µ 04π

I d

v

l × R̂R 2∫

112

dl z2 + a 2

a

z2 + a 2

dl z2 + a 2

z z2 + a 2

dl z2 + a 2

dl cos φ z 2 + a 2

An electron is moving in a straight line with constantspeed v. What approach would you choos e tocalculate the B-field generated by this electron?

A) Biot-SavartB) Ampere’s law (not ‘Maxwell-Ampere’)

C) Either of the above.D) Neither of the above.

e -v

113

3/ 2

2 2 2 3 / 2z=0 plane

( 0, 0, )4

ˆ ˆ( )

×ℜ=

ℜ

+′ ′= −′ ′+ +

∫∫

∫∫

v v

v

o

xy

o o

K B z da

K z y y zdx dy

x y z

µ π

µ π

x

y

z

K

The B-field hasA)y-component onlyB)z-component onlyC)y and z-componentsD)x, y, and z-components

114

We have derived the integral expression for the B-fielda distance z from a current sheet in the z = 0 plane: Stoke’s Theorem: line v. surface integral

Rank order (over blue surfaces)

where J is uniform, going left to right:

A) iii > iv > ii > iB) iii > i > ii > ivC) i > ii > iii > ivD) Something else!!E) Not enough info given!!

r

J • dr

A∫∫ 115 If the arrows represent a B field (notethat | B| is the same everywhere), isthere a nonzero J (perpendicular tothe page) in the dashed region ?

A.YesB.NoC.Need more information to decide

116



14

If the arrows represent a B field (note that |B| is thesame everywhere), is there a nonzero J (perpendicularto the page) in the dashed region?

A.YesB.NoC.Need more information to decide

117

0 ˆB B= ϕv

The magnetic field in a certain region is given by

(C is a positive constant) Consider the imaginary

loop shown. What can you say about the electriccurrent passing through the loop?

A. must be zeroB. must be nonzeroC.Not enough info

118

ˆB(x, y) C yx=v

What is around this purple(dashed) Amperian loop?

r

B • dr

l∫

A) µ0 (|I2 | +|I 1 |) B) µ0 (|I2 |-|I 1|)C) µ0 (| I2 | + | I 1 | sin θ) D) µ0 (| I2 | - | I 1 | sin θ)E) Something else!

119

Stoke’s Theorem: line v. surface integral

Rank order (over blue surfaces), where J isan arbitrary (not necessarily uniform), static currentdensity.

A) iii > iv > ii > iB) iii > i > ii > ivC) i > ii > iii > ivD) Something else!!E) Not enough info given!!

r

J • dr

A∫∫ 120

Pick a sketchshowing B fieldlines thatviolate one ofMaxwell’sequationswithin theregionbounded bydashed lines.

(What currents would be needed to generate the others?)

121

(F)

A solenoid has a total of N windings over a distanceof L meters. We "idealize" by treating this as asurface current running around the surface.What is K?

A) I B) NI C) I/L D) I N/LE) Something else...

122

An infinite solenoid with surface currentdensity K is oriented along the z-axis. ApplyAmpere's Law to the rectangular imaginaryloop in the yz plane shown. What does thistell you about Bz, the z-component of the B-field outside the solenoid?

A) Bz is constant outsideB) Bz is zero outsideC) Bz is not constant outsideD) It tells you nothing about Bz

123z

K

An infinite solenoid with surface currentdensity K is oriented along the z-axis. ApplyAmpere's Law to the rectangular imaginaryloop in the yz plane shown.We can safely assume that B(s ∞ )=0.What does this tell you about the B-fieldoutside the solenoid?

A) |B| is a non-zero constant outsideB) |B| is zero outsideC) |B| is not constant outsideD) We still don’t know anything about |B|

124z

K

A thin toroid (or is it a doughnut?) has (average) radiusR and a total of N windings with current I.We "idealize" this as a surface current running aroundthe surface.What is K, approximately?

A) I/R B) I/(2 π R)C) NI/R D) NI/(2 π R)

E) Something else

125



15

What direction do you expectthe B field to point?

A) AzimuthallyB) RadiallyC) In the z direction

(perp. to page)D) Loops around the rimE) Mix of the above...

126What Amperian loop would youdraw to find B “inside” the Torus(region II)

A) Large “azimuthal” loopB) Small loop in region IIC) Smallish loop from

region II to outside(where B=0)

D) Like A, but perp topage

E) Something entirelydifferent

127

∇2

r

A = − µ 0r

JIn Cartesian coordinates, this means:

, etc.

Does it also mean, in sphericalcoordinates, that ?

∇2A x = − µ 0J x

∇2A r = − µ 0J r

A) YesB) No

128

r

A (r

r ) = µ 04π

r

J (r ' )ℜ

d τ '∫∫∫ Can you calculate that integral usingspherical coordinates?

A) Yes, no problem

B) Yes, r' can be in spherical, but J stillneeds to be in Cartesian componentsC) No.

129

The vector potential A due toa long straight wire withcurrent I along the z-axis is inthe direction parallel to:

z

I

A = ?ˆA) z

ˆB) (azimuthal)ˆC) s (radial)ϕ

130

What is

A) The current density JB) The magnetic field BC) The magnetic flux ΦBD) It's none of the above, but is something

simple and concreteE) It has no particular physical interpretation,

or it is ill defined due to the gauge freedomof A

131

∫ ⋅ ld Ar r

The vector potential in a certain region is given by

(C is a positive constant) Consider the imaginaryloop shown. What can you say about the magneticfield in this region?

A. B is zeroB. B is non-zero, parallel to z-axisC. B is non-zero, parallel to y-axisD. B is non-zero, parallel to x-axis

ˆA(x, y) C yx=v

132

x

yA

z

Choose boundary conditions

Choose all of the following statements

that are implied by(for any closed surface you like) 0=⋅∫∫ ad Br

r

(I)(II)(III)

0=⋅∇ Br r

Babove // = Bbelow

//

⊥⊥ = belowabove B BA) (II) onlyB) (III) onlyC) (I) and (II) onlyD) (I) and (III) onlyE) All of the above

133

In general, which of the following are

continuous as you move past aboundary?

134

A) A B) Not all of A, just Aperp

C) Not all of A, just A //

D) Nothing is guaranteed to becontinuous regarding A



16

The leading term in the vector potentialmultipole expansion involves

What is the magnitude of this integral?

∫ 'lr

d

A) R

B) 2 π R

C) 0

D) Something entirely different/it depends!

135

Two magnetic dipoles m1 and m 2 are oriented inthree different ways.

m1 m21.

2.

3.

Which ways produce adipole field at largedistances?

A) None of theseB) All threeC) 1 onlyD) 1 and 2 onlyE) 1 and 3 only

136

F I L B= ×v v v

B

y

zI (in)

I (out)

x

y

z

B

I

A current-carrying wire loop is in a constant externalmagnetic field B = B z_hat as shown. What is the directionof the torque on the loop?A) Zero B) +x C) +y D) +zE) None of these

137

The force on a segment of wire L is

x

We just found that the torque on a magneticdipole in a B field is:

r

τ =r

m ×r

B

138

How will a small current loop line up ifthe B field points uniformly up the page?(Hint: E-dipoles line up in anti-parallel to E ext)

Griffiths argues that the force on a magneticdipole in a B field is:

r

F =r

∇(r

m •r

B)

If the dipole m points in the z direction,what can you say about B if I tell you theforce is in the x direction?

A) B simply points in the x directionB) Bz must depend on xC) Bz must depend on zD) Bx must depend on xE) Bx must depend on z

139

D) Not enough information to answerE) There is no net force on a dipole

140

K

m

A B C

Suppose I place a small dipole ‘m’ at variouslocations near the end of a large solenoid. At

which point is the magnitude of the force on thedipole greatest?

Which type of magnetic material has thefollowing properties:

1) The atoms of the material have an odd number of electrons2) The induced atomic magnetic dipoles align in the same

direction as an applied magnetic field3) Thermal energy tends to randomize the induced dipoles

A.FerromagneticB.DiamagneticC.Paramagnetic

141 Predict the results of the following

experiment: a paramagnetic bar anda diamagnetic bar are pushed insideof a solenoid.

a) The paramagnet is pushed out, the diamagnet is sucked inb) The diamagnet is pushed out, the paramagnet is sucked inc) Both are sucked in, but with different forced) Both are pushed out, but with different force

142

A solid cylinder has uniform magnetization Mthroughout the volume in the z di rection asshown. Where do bound currents show up?

Α) Everywhere: throughout thevolume and on all surfaces

B) Volume only, not surfaceC) Top/bottom surface onlyD) Side (rounded) surface onlyE) All surfaces, but not volume

143

M



17

A solid cylinder has uniform magnetization Mthroughout the volume in the x direction as

shown. Where do bound currents show up?

144

A) Top/bottom surface onlyΒ) Side (rounded) surface onlyC) EverywhereD) Top/bottom, and parts of

(but not all of) side surface(but not in the volume)

E) Something different/othercombination!

M

A sphere has uniform magnetization M inthe z direction.

A)B)C)D)E) None of these!

M sin θ ϕ̂ M sin θ θ̂

Mcos θ ϕ̂ M cos θ θ̂

Which formula is correct for thissurface current?

145

M

A very long aluminum (paramagnetic!) rodcarries a uniformly distributed current Ialong the +z direction. We know B will be

CCW as viewed from above. (Right?)What about H and M inside the cylinder?

A) Both are CCWB) Both are CWC) H is CCW, but M is CWD) H is CW, M is CCWE) ???

146

A very long aluminum (paramagnetic!)rod carries a uniformly distributedcurrent I along the +z direction.What is the direction of the boundvolume current?

A) J B points parallel to IB) J B points anti-parallel to IC) Other/not sure

147 A very long aluminum (paramagnetic!)rod carries a uniformly distributedcurrent I along the +z direction.What is the direction of the boundsurface current?

A) KB points parallel to IB) KB points anti-parallel to IC) Other/not sure

148 Inside a hollow solenoid,B=B0=µ0nI, ( so H=H 0=nI )If the solenoid is filled with a paramagneticmaterial, what is B inside?...

A) B0B) a little more than B 0

C) a lot more than B 0D) a little less than B 0E) a lot less than B 0

149

Inside a very long hollow solenoid,B=B0=µ0nI, ( so H=H 0=nI )If the solenoid is filled with iron,what is H inside?...

A) H0B) a little more than H 0C) a lot more than H 0D) a little less than H 0E) a lot less than H 0

150A large chunk of paramagnetic material ( χ m>0)

has a uniform field B0 throughout its interior.We cut out a cylindrical hole (very skinny, verytall!)

What is B at the center of that hole?A) B0 B) more than B 0 C) less than B 0D) ??

151A large chunk of paramagnetic material ( χ m>0)has a uniform field B

0throughout its interior.

We cut out a wafer-like hole (very wide, veryshort!)

What is B at the center of that hole?A) B0B) more than B 0C) less than B 0D) ??

152



18

A sphere (with a spherical cavity inside it) is madeof a material with very large positive χm.It is placed in a region of uniform B field.Which figure best shows the resulting B field lines?

E) None of these can be even remotely correct

153

1: C 11: B 21: C 31: D 41: A 51: B2: D 12: B 22: C 32: A 42: A 52: A

3: B 13: C 23: C 33: C 43: B 53: B

4: B 14: D 24: C 34: A 44: B 54: C

5: A 15: D 25: B 35: B 45: B 55: A

6: A 16: E 26: B 36: C 46: B 56: A

7: D 17: E 27: A 37: D 47: B 57: A

8: B 18: C 28: C 38: D 44: B 58: A

9: D 19: E 29: A 39: A 49: B 59: E

10: D 20: A 30: B 30: B 50: B 60: E

Answers

61: D 71: B 81: A 91: A 101: C 111: D62: C 72: E 82: C 92: D 102: B 112: A

63: C 73: E 83: B 93: A 103: D 113: D

64: D 74: B 84: B 94: A 104: A 114: A

65: D 75: C 85: A 95: E 105: A 115: D

66: D 76: B 86: A 96: C 106: D 116: A

67: D 77: B 87: A 97: A 107: C 117: A

68: B 78: D 88: B 98: A 104: C 118: B

69: C 79: B 89: D 99: C 109: C 119: A

70: E 80: A 90: C 100: A 110: D 120: D

Answers

121: D 131: C 141: C 151: C

122: D 132: B 142: B 152: A

123: A 133: D 143: D 153: A

124: B 134: A 144: D

125: D 135: C 145: B

126: A 136: E 146: A

127: A 137: B 147: A

128: B 138: B 148: B129: B 139: B 149: B

130: A 140: B 150: A

Answers

Electromagnetism. KU. Concept Test Practice

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