Scalar Product:

cos AB x x y y z zA B A B A B A B A B

Vector Product:

sin

ˆ

ˆ

ˆ

AB

y z z y

z x x z

x y y x

A B A B

A B A B i

A B A B j

A B A B k

Equations of motion:

0

2

0 0

2 2

0 0

1

2

2 ( )

v v at

x x v t at

v v a x x

Radial Acceleration: 2

rad

va

r

Newton’s second law

amF

Magnitude of kinetic friction

Nkf FFk

Magnitude of static friction

sf s NF F

Definition of work

xdFW

Definition of kinetic energy:

21

2KE mv

Change in gravitational potential energy:

gU mg y

Elastic potential energy:

21

2elU kx

Work-Energy Theorem:

W U KE

Center-of-mass position

n

i

iiCOM mxM

X1

1

Definition of momentum

vmp

Conservation of momentum

fi pp

Definition of torque

Fr

Newton’s second law for rotation

I

Conditions for rolling:

and COM COMa R v R

Angular momentum:2or , where i i

i

L r p L I I m r

Newton’s Law of Gravitation:

1 2

2G

Gm mF

r and 1 2

G

Gm mU

r with

0GU at infinity

Bernoulli's Equation:

2 2

1 1 1 2 2 2

1 1

2 2p gy v p gy v

Equation for Simple Harmonic Motion: 2

2

2

d xx

dt

Solution for above equation:

cosx t A t

Where,

22 f

T

For a spring mass oscillator,

k

m

For a simple pendulum,

g

L

Wave Equation:

2 2

2 2 2

, ,1y x t y x t

x v t

Solution to above equation:

, cosy x t A kx t

Where, 2

, 2 ,k f v f

Standing waves on fixed string:

, sin sin

2

SW

n

y x t A kx t

vf n

L

Doppler Effect:

LL S

S

v vf f

v v

5

329

o oT C T F

273.15oT K T C

0

0

L L T

V V T

F V F V

Q mc T nC T

Q mL

H C

dQ AH k T T

dt L

pV nRT

3

2trK nRT

3

2

5

2

V

V

C R

C R

2

1

V

V

W pdV

U Q W

for ideal gasVU nC T

1 adiabatic process

pV const

TV const

1 C

H H

W Qe

Q Q

1 CCarnot

H

Te

T

2

1

dQS

T

lnS k w

8.314R J mol K

236.02 10AN molecules mole

1 atm = 101 325 N / (m2) = 1.01x10

5 Pa

1/4πε0 = 8.99 x 109 Nm

2/C

2

e = -1.602 x 10-19

C

2

0

1ˆ

4E

qQF qE r

r

0

encE

qE dA

b

b a

a

V V E dl

U q V

ˆˆ ˆV V VE i j k

x y z

Q CV

1 2 3

1 1 1 1series

eqC C C C

1 2 3 paralleleqC C C C

21

2U CV

2

0

1

2Eu E

0EE

K

dqI

dt

dJ nqv

E

J

LR

A

V IR

P VI

1 2 3 serieseqR R R R

1 2 3

1 1 1 1parallel

eqR R R R

1 chargingtRCq C e

0 dischargingtRCq Q e

F qv B

B B dA

dF Idl B

B , NIA

ideal monatomic gas

ideal diatomic gas w/o vibration

U B

0

2

ˆ

4

qv rB

r

0

2

ˆ

4

Idl rdB

r

0 encB dl I

0 C DB dl i i

Bddl

dt

ED

di

dt

1 22 1

2 2 1 1

1 2

and

B B

di diM M

dt dt

N NM

i i

,

B

diL

dt

NL

i

21

2U LI

,

2

0

1

2Eu B

Rt

Ldi

edt L

1

LC

1for cos

2

1for cos

2

RMS

RMS

I I i I t

V V v V t

22

, where

1, where

, where

R

L L L

C C C

L C

V IR

V IX X L

V IX XC

V IZ Z R X X

Avg

1cos , tan

2

L CX XP VI

R

ss p

p

NV V

N

Physics 161-001 Spring 2014 Exam 3 Name: __________________________________ Box#_____________

Multiple Choice (5 points each):

1) A tube of mercury with resistivity 7.84 × 10-6

Ωm has an electric field inside the

column of mercury of magnitude 8 V/m that is directed along the length of the

tube. How much current is flowing through this tube if its radius is 6.0 mm?

A) 4.80 A

B) 6.00 A

C) 10.0 A

D) 12.3 A

E) 19.2 A

F) 25.0 A

G) 55.4 A

H) 87.3 A

I) 115 A

J) 134 A

2) The emf and the internal resistance of a battery are as shown in the figure. When

the terminal voltage Vab is equal to 21.0 V, what is the current through the

battery?

A) 1.2 A

B) 4.3 A

C) 5.0 A

D) 10.0 A

E) 15.5 A

F) 19.1 A

G) 23.2 A

H) 26.0 A

I) 52.0 A

J) 104 A

6 2

6

81.0 10

7.84 10

E V mJ A m

m

and

2

6 2 31.0 10 6 10 115I J A A m m A .

21.0 31 2

5.0

abV V Ir V I

I A

E

3) A proton moving in the positive x direction enters a magnetic field. If the proton

experiences a magnetic deflection in the negative y direction, the magnetic field

in this region is

A) in the direction of the +x axis.

B) in the direction of the -x axis.

C) in the direction of the +y axis.

D) in the direction of the -y axis.

E) in the direction of the +z axis.

F) in the direction of the -z axis.

G) in any direction perpendicular to the proton velocity.

H) zero.

I) undefined.

4) If the current density in a wire of radius R is given by J = J0 + kr, 0 < r < R, what

is the total current in the wire?

A) kR2/2

B) kR2

C) J0R + kR2/2

D) J0R2 + kR

3/3

E) J0πR2 + k2πR

3/3

F) J02πR + k2πR2/2

G) J0πR + k2R2/2

H) k2πR3/3

I) kR3/3

J) J0(k2πR3/3)

BF qv B . Since the proton is positively charged,

the direction is in the same direction given by the right

hand rule.

0 0

0 0

0 0 0

2

0

0 0

2 3

0

2

2 2 2

2 2

2 3

R R

R R R

R R

I J r dA J r rdr

J kr rdr J rdr kr rdr

J rdr k r dr

J R k R

5) The figure shows the cross-section of a hollow cylinder of inner radius a = 1.0 cm

and outer radius b = 2.0 cm. A uniform current density of 1.0 A/ cm2 flows

through the cylinder parallel to its axis. Calculate the magnitude of the magnetic

field at a distance of d = 1.0 m from the axis of the cylinder. (μ 0 = 4π × 10-7

T ·

m/A)

A) 1.1 x 10

-5 T

B) 1.5 x 10-5

T

C) 1.1 x 10-6

T

D) 1.9 x 10-6

T

E) 6.3 x 10-6

T

F) 9.7 x 10-6

T

G) 2.2 x 10-7

T

H) 4.8 x 10-7

T

I) 9.1 x 10-7

T

J) 0 T

6) Calculate the current through a 1.0-m long 22 gauge (having radius 0.321 mm)

nichrome wire if it is connected to a 3.0-V battery. The resistivity of nichrome is

100 × 10-8

Ω · m.

A) 1 A

B) 2 A

C) 3 A

D) 4 A

E) 5 A

F) 6 A

G) 7 A

H) 8 A

I) 9 A

J) 10 A

In this case, J is uniform, so,

2 2 2 2 2

2 2 2

1

1 4 1 9.4

I JA J b a A cm b a

A cm cm cm A

Then, from symmetry, we use Ampere’s law to find

that:

0

7

6

2 4 10 9.4

1.9 10

encB d s i

T mB r AA

B T

8

24

100 10 13.1

3.21 10

L m mR

A m

and,

31

3.1

V IR

V VI A

R

7) The figure shows three identical light bulbs connected to a battery having a

constant voltage across its terminals. What happens to the brightness of light bulb

3 when the switch S is closed?

A) Momentarily goes up then back to its original brightness.

B) Momentarily goes down then back to its original brightness.

C) Permanently gets brighter.

D) Permanently gets dimmer.

E) No change.

8) A current is running through a wire next to the circuit shown in the figure with the

switch S open and the capacitor uncharged. The battery has no appreciable

internal resistance. Which one of the following graphs best describes the

magnitude of the force on the wire as a function of time t after closing the switch?

F

F

F

F

F

A

C

B

D

E

9) For the circuit shown in the figure, determine the current in the 4.0-Ω resistor.

A) 0.1 A

B) 0.2 A

C) 0.4 A

D) 0.6 A

E) 0.8 A

F) 1.0 A

G) 1.2 A

H) 1.4 A

I) 1.6 A

J) 1.8 A

10) Consider the circuit shown in the figure. Note that two currents are shown.

Calculate the emf ε3.

A) 50 V

B) 48 V

C) 44 V

D) 42 V

E) 40 V

F) 38 V

G) 36 V

H) 34 V

I) 32 V

J) 30 V

We first find the equivalent resistance of the three resistors in parallel:

1 1 1 1 4

2 2 2 5eq

eq

RR

, then the equivalent resistance of

the three in series: 4

2 1.2 45

eqR

and, then the current through the entire circuit: I = V/R = 3A. Then, the

voltage across the resistors in parallel is 12/5V and so the current

through the 4.0-Ω resistor is 3/5 A.

31

3.1

V IR

V VI A

R

2.0

2.0

1.2

21.0 31 2

5.0

abV V Ir V I

I A

E

10.0

2.0 A

11) For the circuit shown in the figure, the switch S is initially open and the capacitor

voltage is 80 V. The switch is then closed at time t = 0. What is the charge on the

capacitor when the current in the circuit is 13 μA?

A) 110 μC

B) 140 μC

C) 200 μC

D) 220 μC

E) 280 μC

F) 330 μC

G) 390 μC

H) 430 μC

I) 470 μC

J) 500 μC

12) For the circuit shown in the figure, the capacitors are all initially uncharged, the

connecting leads have no resistance, the battery has no appreciable internal

resistance, and the switch S is originally open. Just after closing the switch S,

what is the current in the 15.0-Ω resistor?

A) 0 A

B) 0.2 A

C) 0.4 A

D) 0.6 A

E) 0.8 A

F) 1.0 A

G) 1.2 A

H) 1.4A

I) 1.6 A

J) 1.8 A

0

t RCq Q e and 0 t RCdq Q

i edt RC

.

Now, at some time, i = 13μA, or:

0

0 0 0

13

13 13 13

t RC

t RC

Qi e A

RC

RC RC Re A A A

Q CV V

then,

0 0

0

13 13 430t RC Rq Q e Q A C R A C

V

Immediately after the switch is closed, the voltage

drop across the capacitor is 0V, so no current

through the 15 Ω resistor.

30.0

13) After the switch S has been closed for a very long time, what is the potential

difference across the 28.0-μF capacitor?

A) 0.0 V

B) 4.3 V

C) 5.0 V

D) 10.0 V

E) 15.0 V

F) 19.1 V

G) 23.2 V

H) 26.0 V

I) 30.0 V

J) 104 V

14) A charge is accelerated from rest through a potential difference V and then enters

a uniform magnetic field oriented perpendicular to its path. The field deflects the

particle into a circular arc of radius R. If the accelerating potential is doubled to

2V, what will be the radius of the circular arc?

A) 2R

B) R/2

C) 4R

D) R/4

E) 2 R

F) R / 2

G) R

H) 3R

I) 3 R

J) R/ 3

21

2

2

KE mv qV

qvv

m

and

2

2 2 1 2

BF qv B

vqvB ma m

R

mv mv m qV mVR

qvB qB qB m B q

For long times, no current passes through the

capacitors, so the current just goes through the three

resistors in series. The equivalent resistance is just

30 Ω, with 30V means 1 A is passing through each

resistor. Then the voltage drop across the 15 Ω

resistor is just 15 V.

30.0

15) A circular coil of wire of 100 turns and diameter 10.0 cm carries a current of 1.0

A. It is placed in a magnetic field of 1 T with the plane of the coil making an

angle of 45° with the magnetic field. What is the magnetic torque on the coil?

A) 2.0 Nm

B) 2.2 Nm

C) 2.4 Nm

D) 2.6 Nm

E) 2.8 Nm

F) 3.0 Nm

G) 3.2 Nm

H) 3.4 Nm

I) 3.6 Nm

J) 3.8 Nm

16) A negatively charged particle is moving to the right, directly above a wire having

a current flowing to the right, as shown in the figure. In which direction is the

magnetic force exerted on the particle?

A) up

B) down

C) out of the page

D) into the page

E) no magnetic force

B and

2 2100 1.0 0.1 3.14NIA A m Am so

2sin 3.14 1 sin45 2.22B Am T Nm

The B-field from the wire is coming out of the page

at the charge, and from F qv B , with q

negative, then the force is upwards.

17) A large number of very long wires of diameter 1mm are laid side-by-side to form

a plane. If 10.5 A of current is passed through each wire (in the same direction),

what is the magnitude of the magnetic field 10cm above (and in the middle of) the

plane? (μ 0 = 4π × 10-7

T · m/A)

A) 1.6 x 10-4

T

B) 5.2 x 10-4

T

C) 7.9 x 10-4

T

D) 2.6 x 10-3

T

E) 4.7 x 10-3

T

F) 6.6 x 10-3

T

G) 9.2 x 10-3

T

H) 2.1 x 10-2

T

I) 4.4 x 10-2

T

J) 6.5 x 10-2

T

0

7

3

7

3

3

10.52 4 10

1 10

10.52 10

1 10

6.6 10

encB d s i

AT mBL LA m

AT mBA m

T