GENERAL PHYSICS PH 222-2A (MIROV)
Exam 3 (03/31/15)
STUDENT NAME: _______________________STUDENT id #: ______________________
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WORK ONLY 5 QUESTIONS
ALL QUESTIONS ARE WORTH 30 POINTS
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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
1.
Magnetic force exerted on a point charge by a magnetic field: F

B
 q v



B
2.
The number density n of charge carriers (Hall Effect): n

B i
V l e
3.
Circular orbit in magnetic field: r
 m v q B
4.
Magnetic force on a straight current carrying wire of length L:
5.
The force acting on a current element in a magnetic field: d F

B

6.
7.
Magnetic dipole moment of current loop:
Torque on a current carrying coil:


 




B
= [current] x [area]
8.
Orientation energy of a magnetic dipole: U (

)
  



B
F

B  i d L


 i L

B


B
9.
The work done on the dipole by the agent is:
10.
Permeability constant:
 o
= 1.26x10
-6
N
 s
2
/C
2
= 1.26x10
-6
H/m;
11.
Permittivity constant:
 o
= 8.85x10
-12
C
2
/(N
W a
 
U

U f

U i
 m
2
) = 8.85x10
-12
F/m;
1.
The Bio-Savart Law:
 d B

4

Ch. 29 Magnetic fields due to currents o i d s

 r ˆ
 r
2
2.
Magnetic field of a long straight wire current: B


2

0
I r
{Unit 1 tesla = 1T = 1 N/(C

m/s)}
3.
Magnetic field of a circular arc: B

4
 o i
R
4.
Forces between parallel currents: F b a
 a s i n 9 0


2 o
5.
Ampere’s Law:


B
 d s

  o i e n c
L i i b
 d
B
 
6.
Magnetic field of an ideal solenoid:
7.
Magnetic field of a toroid: B

 o i N
2

1 r o i n

1.
Field of a magnetic dipole:

B ( z )

2
 o

 z

3
Ch. 30. Induction and Inductance
1.
Magnetic flux:

B
 

B

 d A {Unit 1weber = 1Wb = 1 T m
2
}
2.
Faraday’s Law of Induction:
   d

B d t
3.
Lenz’ Law: Induced emf opposes change that produced it
4.
Emf and the Induced Electric Field:
5.
Inductance: L

N
 i
B
  
E

 d s

  d
 d t
B
{SI unit henry(H), where 1 H=1Tm
2
/A)
6.
The inductance per unit length of solenoid:
L l
  o n 2 A
7.
Self induction:
  
L
L d i
8.
Mutual induction: M
9.
Series RL Circuits: i

 d t


R i
1
B
2
( 1

 e
 i
2
 t

L
B
1

2
 
M d i d t
  
M d i d t
) ( r i s e o f c u r r e n t ) i

 t

L ( d e c a y o f c u r r e n t )
10.
Magnetic Energy:
U u
B
B


1
L i 2
2
2
B

2 o
( m a g n e t i c e n e r g y )
( m a g n e t i c e n e r g y d e n s i t y )
Ch. 31. Electromagnetic Oscillations and Alternating Current
1.
LC Energy transfer: U
E

2 q 2
C
U

L i
2
2
U

U
E
2.
Emf of an electromagnetic generator:


U
B
 c o n s t
= -

/
 t=NAB

sin(

t)
3.
LC Charge and Current Oscillations: q

Q c o s (
 t
   
1
L C i
  
Q s i n (

4.
Damped Oscillations: q

Q e

R t 2 L c o s (

' t
 
) , w h e r e

'
 
2  
R 2 L

2
5.
Alternating Currents; Forced Oscillations:
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 r f r e q u e n c y
 d
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

=
 m s i n
 d t ; i

I s i n (
 d t
 
) t
 
)

1.
Resonance: I

 m w h e n
 d
 
. T h e n X
C

X
L
,
 
0
R
2.
Capacitive reactance: V
C
=IX
C
; X
C


1 d
C
; the current leads voltage by

/2 radians (

=-

/2 rad)
3.
Inductive reactance: V
L
=IX
L
; radians(

=

/2 rad)
X
L
  d
L ; the current lags behind the voltage by

/2
4.
Series RLC Circuits
Relation between emf and current:
 m
=IZ
Impedance: Z

R
2 

X
L

X
C

2
Phase angle between current and voltage: tan
 
X
L

R
X
C
Average Power dissipated:
P a v g

I
2 r m s
R
  r m s
I r m s c o s

I

I / V
Power factor of the circuit: cos


V /
  
/ 2
V s

V p
N
N s p
( t r a n s f o r m a t i o n o f v o l t a g e )
5.
Transformers:
I s

I p
N p
( t r a n s f o r m a t i o n o f c u r r e n t s )
N s
T h e e q u i v a l e n t r e s i s t a n c e o f t h 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 t h e g e n e r a t o r
R e q 
N
N s p


2
R , w h e r e R i s t h e r e s i s t i v e l o a d o f t h 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:

B
 
2.
Maxwell’s Extension of Ampere’s Law:

B


 d A

B

 d S

0
  
0
3.
Displacement current: i d
 
0 d
 d t
E d
 d t
 
0 i e n c

1.
The rectangular loop shown in Figure is pivoted about y axis and carries a current of 15.0 A in the direction indicated. a) Find magnitude and direction of the loop magnetic dipole moment. b) If the loop is in uniform magnetic field with magnitude 0.48 T in the +x direction, find the magnitude and direction of the torque required to hold the loop in the position shown. c) What is the direction in which the loop tends to rotate?

2.
A circular loop has radius R and carries current I
2 in a clockwise direction.
The center of the loop is a distance D above a long straight wire. What are the magnitude and direction of the current I
1
in the wire if the magnetic field at the center of the loop is zero?

3.
current whose current density is J . The current density, although symmetric about the cylinder axis, is not constant but varies according to relationship
J



2 I a o
2
 


1


 r
 a



2
 

 k

f o r r
 a

0 f o r r
 a
Where a is the radius of the cylinder, r is the radial distance from the the cylinder axis, and I o
is a constant having units of amperes. a) Show that I o
is the total current passing through the entire cross-section of the wire. b) Obtain the expression of the current I contained in a circular cross-section of radius r
 a and centered at the cylinder axis. c) Using Ampere’s law, derive an expression for the magnitude of the magnetic field inside in the region r
 a .

4.
The drawing shows a plot of the output emf of a generator as a function of time t .
The coil of this device has a cross-sectional area per turn of 0.020 m
2
and contains
150 turns. Find: a) The frequency f of the generator in hertz; b) The angular speed

in rad/s; and c) The magnitude of the magnetic field.

5.
A friend returns to the United States from Europe with a 960-W coffeemaker, designed to operate from 240-V line.
A) Determine the ration N s
/N p
of the transformer necessary to operate the coffeemaker in the United States.
B) What current will the coffeemaker draw from the 120-V line?
C) What is the resistance of the coffeemaker? (The voltages are rms values.)

7.
An oscillating LC circuit has current amplitude of 7.50 mA, potential amplitude of 250 mV, and a capacitance of 220 nF. What are: a) The period of oscillation; b) The maximum energy stored in the capacitor; c) The maximum energy stored in the inductor; d) The maximum rate at which the current changes; and e) The maximum rate at which the inductor gains energy.
(a) From V = IX
C
we find

= I/CV . The period is then T = 2

/

= 2

CV/I = 46.1
 s.
(b) The maximum energy stored in the capacitor is
U
E

1
2
C V
2 
1
2
( 2 .2 0

1 0

7
F )( 0 .2 5 0 V )
2 
6 .8 8

1 0

9
J .
(c) The maximum energy stored in the inductor is also U
B

L I
2
/ 2

6.88 nJ .
(d) V = L ( di/dt ) max
. We can substitute L = CV
2
/I
2
and solve for ( di/dt ) max
. Our result is

 d i d t

 m a x

V

V
L C V
2
/ I
2

I
2
C V

( 7 .5 0

1 0

3
A )
2
( 2 .2 0

1 0

7
F )( 0 .2 5 0 V )

1 .0 2

1 0 A /s .
(e) The derivative of U
B
=
1
2
Li
2
leads to d U
B d t

L I
2
 s in
 t c o s
 t
Therefore, d U
B d t m a x

1
2
L I
2
 
1
2
IV


1
2
L I
2
 s in 2
 t .
1
2
( 7 .5 0

1 0

3
A )( 0 .2 5 0 V )

0 .9 3 8 m W .