COLLEGE PHYSICS PH 222-3A (MIROV) Exam 3 (04/04/11)

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ALL QUESTIONS ARE WORTH 30 POINTS

------------------------------------------------------------------------------------------------------------------------------------------NOTE: Clearly write out solutions and answers (circle the answers) by section for each part (a., b., c., etc.)

Important Formulas:

Ch 28 Magnetic fields

WORK ONLY 5 QUESTIONS

Ch.28. Magnetic fields 1. Magnetic force exerted on a point charge by a magnetic field: BF q v B

2. The number density n of charge carriers (Hall Effect): B inV l e

3. Circular orbit in magnetic field: m vrq B

4. Magnetic force on a straight current carrying wire of length L: BF i L B

5. The force acting on a current element in a magnetic field: Bd F i d L B

6. Magnetic dipole moment of current loop: = [current] x [area] 7. Torque on a current carrying coil: B

8. Orientation energy of a magnetic dipole: ( )U B

9 The work done on the dipole by the agent is: W U U U9. The work done on the dipole by the agent is: a f iW U U U

10. Permeability constant: o = 1.26x10-6 Ns2/C2 = 1.26x10-6 H/m; 11. Permittivity constant: o = 8.85x10-12 C2/(Nm2) = 8.85x10-12 F/m;

Ch. 29 Magnetic fields due to currents

1. The Bio-Savart Law: 2

ˆ4

o i d s rd Br

4 r

2. Magnetic field of a long straight wire current: B Ir

0

2 {Unit 1 tesla = 1T = 1 N/(Cm/s)}

3. Magnetic field of a circular arc: 4

o iBR

4. Forces between parallel currents: s i n 9 02o a b

b a b aL i iF i L B

d

2 d5. Ampere’s Law: o e n cB d s i

6. Magnetic field of an ideal solenoid: oB i n

7. Magnetic field of a toroid: 12

o i NBr

1. Field of a magnetic dipole: 3( )2

oB zz

Ch. 30. Induction and Inductance

1. Magnetic flux: B B d A

{Unit 1weber = 1Wb = 1 T m2}1. Magnetic flux: B B d A {Unit 1weber 1Wb 1 T m }

2. Faraday’s Law of Induction: Bdd t

3. Lenz’ Law: Induced emf opposes change that produced it

4. Emf and the Induced Electric Field: BdE d sd t

BN 25. Inductance: BNLi

{SI unit henry(H), where 1 H=1Tm2/A)

6. The inductance per unit length of solenoid: 2o

L n Al

7. Self-induction: Ld iLd t

8 M t l i d ti 2 1 1 2B B d i d iM M M

8. Mutual induction: 2 1 1 22 1

1 2

; ; B B d i d iM M Mi i d t d t

9. Series RL Circuits: ( 1 ) ( r i s e o f c u r r e n t )

( d e c a y o f c u r r e n t )

L

L

t

t

o

i eR

i i e

21 ( m a g n e t i c e n e r g y )U L i10. Magnetic Energy: 2

( m a g n e t i c e n e r g y )2

( m a g n e t i c e n e r g y d e n s i t y )2

B

Bo

U L i

Bu

Ch. 31. Electromagnetic Oscillations and Alternating Current

2 2

1. LC Energy transfer: 2 2

, , 2 2E B E Bq L iU U U U U c o n s t

C

2. Emf of an electromagnetic generator: = - /t=NAB sin( t)

3. LC Charge and Current Oscillations: 1c o s ( ) ; ; s i n ( )q Q t i Q tL C

4 Damped Oscillations: 22 2c o s ( ' ) w h e r e ' 2R t Lq Q e t R L 4. Damped Oscillations: c o s ( ) , w h e r e 2q Q e t R L 5. Alternating Currents; Forced Oscillations:

d m

A s e r i e s R L C c i r c u i t m a y b e s e t i n t o f o r c e d o s c i l l a t i o n a t a d r i v i n g a n g u l a rf r e q u e n c y b y a n e x t e r n a l a l t e r n a t i n g e m f = s i n ; s i n ( )d dt i I t

1. Resonance: w h e n . T h e n , 0md C LI X X

R

2. Capacitive reactance: VC=IXC; 1C

d

XC

; the current leads voltage by /2 radians (=-

/2 rad)/2 rad) 3. Inductive reactance: VL=IXL; L dX L ; the current lags behind the voltage by /2

radians(=/2 rad) 4. Series RLC Circuits

Relation between emf and current: m=IZ

22Impedance: 22CL XXRZ

Phase angle between current and voltage: R

XX CL tan

Average Power dissipated: 2 c o s , / 2 ; / 2 ; / 2a v g r m s r m s r m s r m s r m s r m sP I R I I I V V a v g r m s r m s r m s r m s r m s r m s

Power factor of the circuit: cos

( t r a n s f o r m a t io n o f v o l ta g e )ss p

p

NV VN

5. Transformers:

2

( t r a n s f o r m a t io n o f c u r r e n ts )

T h e e q u iv a le n t r e s i s ta n c e o f th e s e c o n d a r y c i r c u i t , a s s e e n b y th e g e n e r a to r

h i h i i l d

p

ps p

s

p

NI I

N

N

f h d i i

, w h e r e R i s th e r e s i s t iv e lo a d ope q

s

R RN

f th e s e c o n d a r y c i r c u i t

Ch. 32. Maxwell’s Equations. Magnetism of matter

1 Gauss’ Law for Magnetic Field: 0B d A

1. Gauss Law for Magnetic Field: 0B B d A

2. Maxwell’s Extension of Ampere’s Law: 0 0 0 e n cdB d S id t

3. Displacement current: 0E

ddi

d t

1.

4

2. The current density inside a long, solid, cylindrical wire of radius a=3.1 mm is in the direction of the central axis, and its magnitude varies linearly with radial distance r from the axis according to , where Jo=310 A/m^2 . Find the magnitude of the magnetic field at a) r=0;

0rJ Ja

a) r=0;b) r=a/2;c) r=a

For ,r r

r A

i

0 02

( ) ( )2 22 2 2

r ro enc o o

o

o o

i rB r J r rdr J rdrr r r a

J r

.3

(a) At 0, 0.(b) At / 2, we have

ar Br a

7 2 -2 27

2

(4 10 Tm/A)(310A/m )(3.1 10 / 2)( ) 1 10 .3 3(3.1 10 )o enci mB r Ta m

(c) At ,

( )

r a

B r

7 2 -2 2

72

(4 10 Tm/A)(310A/m )(3.1 10 ) 4.0 10 .3 3(3 1 10 )o enci m Ta m

3 3(3.1 10 )a m

3.

4.

7

5.

8

6. A capacitor in an LC oscillator has a maximum potential difference of 15V and a maximum energy of 360 μJ. At a certain instant the energy in the capacitor is 40 μJ. At that instant what is the emf induced in the inductor?

2 62UC 2 6,max 6max

,max 2 2max

21

1 1

2 2 360 101) ; 3.2 10 3.22 15

2) At a certain instant t ; potential difference across the capacitor

EE

E

UCVU C F FV

CVU

1 1

61

1 6

) ; p p2

2 2 40 10 5.03.2 10

E

EUV VC

3) At any instant 2 2

in an oscillating LC circuit 2 2

Since remains constant with time

B ELi qU U U

CU

2 2

1 1

0;2 2

0; at 5.0L L

dU d Li q di q dq diLi Li Vidt dt C dt C dt dt

diL V V t V V

1 1 0; at 5.0L LL V V t V Vdt

7.

V VR

I

VL R

=‐28 Vo

VC‐VL

VC

arccos 0.88 28deg

8. A 1-μF capacitor ( oACd

) is connected to an emf that is increasing uniformly with time at a

rate of 100V/s. What is the displacement current between the plates of the capacitor?

V

0 0 0 0

( )( )Ed

Vd Ad d EA A dV dVdi Cdt dt dt d dt dt

6 4 (1 10 ) (100 ) 1 10d

dt dt dt d dt dtVi F As

3. Immediately after switch S in the circuit shown is closed, what is the current through the battery?

9.

The inductor prevents a fast build-up of the current through it,

0

so immediately after the switch is closes, the current in the inductor is zero.

It follows that ViR R

1 2R R