Physics111 Exam1 Solutions W10 - Union Collegeminerva.union.edu/.../Physics111_Exam1_Solutions_W10.pdfMicrosoft Word - Physics111_Exam1_Solutions_W10.docx Author labrakes Created Date

Physics 111

Winter 2010

Exam #1

January 25, 2010

Name______________

In keeping with the Union College policy on academic honesty, it is assumed that you will neither accept nor provide unauthorized assistance in the completion of this work.

Part Multiple Choice / 12

Problem #1 / 32 Problem #2 / 32 Problem #3 / 24

Total / 100

Part I: Free Response Problems Please show all work in order to receive partial credit. If your solutions are illegible

no credit will be given. Please use the back of the page if necessary, but number the problem you are working on.

1. You are given the collection of point charges shown below. The relevant data about

the charges are given in the table below. Charge Number Charge (µC) x-position (m) y-position (m)

q1 -5 0 3 q2 2 4 0 q3 -6 -2 0

a. What is the net electric field at the point (0,0)?

€

Enet,x = −E2 − E3 = −kq2r22 −

kq3r32 = −9 ×109 Nm 2

C 2

2 ×10−6C4m( )2

+6 ×10−6C2m( )2

= −14625 NC

Enet,y = +E1 = +kq1r12 = 9 ×109 Nm 2

C 2

5 ×10−6C3m( )2

= +5000 NC

∴Enet = Enet ,x2 + Enet,y

2 @φ = tan−1Enet ,y

Enet ,x

→ Enet =15456 N

[email protected]

The direction is given with respect to the negative x-axis. So this is 18.9o above the negative x-axis.

b. What is the electric potential at the point (0,0)?

€

Vnet =V1 +V2 +V3 =kq1r1

+kq2r2

+kq3r3

Vnet = 9 ×109 Nm 2

C 2

−5 ×10−6C3m

+2 ×10−6C4m

+−6 ×10−6C

2m

= −3.75 ×104V = −37.5kV

c. How much work is done moving a charge of q4 = -3 µC from very far away to the

point (0,0)?

€

Wq4= q4ΔVi, f = −3×10−6C × 0 − −37500V( )[ ] = −0.113J = −113mJ

d. What is the net electrostatic force on q4 when it is at the point (0,0)?

€

F→

net,4 = q4 E→

net = 3×10−6C ×15456 NC = 0.046N = 46mN directed at 18.9o below the

positive x-axis since this is a negative charge it feels a force directed opposite the field direction.

Beam enters Beam exits

0V 0V 1.0MV

H- H- H+ H+

2.0m

Eleft Eright

Fleft Fright

2. A one-megavolt tandem electrostatic particle accelerator is designed to accelerate protons (hydrogen nuclei) from rest to a specific energy in two steps. The first stage involves adding an extra electron to a neutral hydrogen atom and accelerating this through a 1.0MV difference in potential as shown on the left side of the diagram below. The second stage involves stripping off all of the electrons at the midpoint of the accelerator and accelerating the proton through another 1.0MV as shown on the right side of the diagram below.

a. How much work is done on the proton from the moment the beam enters the accelerator until the beam exits? (Hint: You’ll need to calculate the work done in two parts, the first is over the left hand accelerating region from 0V to 1MV and the second is over the right hand accelerating region from 1MV to 0V.) Express your answer in MeV and Joules. Using the work done, what is the speed of the proton when it exits the accelerator assuming that it is very nearly at rest when it enters?

€

Wnet =Wleft +Wright = qΔVleft + qΔVright = −e 0 −1MV( ) + e 1MV − 0( ) = 2MeV ×1.6 ×10−19J

1eV= 3.2 ×10−13J

€

W = ΔKE = KE f = 12mv f

2 → v f =2Wm

=2 × 3.2 ×10−13J1.67 ×10−27kg

=1.96 ×107 ms

b. Suppose that after the proton is accelerated it is directed at a target made of lead? What

is the distance of closest approach if the collision is assumed to be elastic and “head-on?” (Hint: Assume that the lead atom does not recoil and the chemical information for lead is:

€

82209Pb .)

€

ΔKE + ΔEPE = KE f −KEi( ) + EPE f − EPEi( ) = −KEi + EPE f = −W + EPE f = 0

∴W = EPE f =ke 82e( )

r→ r =

82ke2

W=

82 × 9 ×109 Nm 2

C 2 × e2 × 1.6 ×10−19C1e

2

3.2 ×10−13J= 5.9 ×10−14m

c. If the accelerating region has a total length of 2.0m, what is the electric field on either

side of the midpoint (where the potential is1.0mV)? What is the force on the proton in each accelerating regions? Draw these vectors on the diagram above.

€

Eleft =ΔVi, f

Δx=0 −1MV1m

= −1MVm → Fleft = −eEleft =1MeVm

×1.6 ×10−19J

1eV=1.6 ×10−13N and

€

Eright =ΔVi, f

Δx=1MV − 01m

=1MVm → Fright = eEright =1MeVm

×1.6 ×10−19J

1eV=1.6 ×10−13N

d. Using the idea of forces from part c, what is the speed of the proton when it leaves the

accelerator?

€

W = FΔx = ΔKE = 12mv f

2 → v f =2FΔxm

=2 ×1.6 ×10−13J × 2m1.67 ×10−27kg

=1.96 ×107 ms

y = 30e-0.01x

0 5

10 15 20 25 30 35

0 50 100 150 200

Volta

ge (V

)

time (s)

A 10000µF capacitor discharging through an unknown resitance R as a function of time.

3. Suppose that you have the graph of the potential difference (in Volts) across a discharging 10,000 µF capacitor as a function of time (in seconds) through an unknown resistor as shown below.

a. What is the value of the unknown resistance?

From the graph we see that

€

1RC

= 0.01stherefore

€

τ = RC→ R =τC

=1

0.01s×

10.01F

=1×104Ω =10kΩ .

b. How much charges was initially stored on the capacitor and what was the initial current that flowed through the resistor?

€

Q = CV = 0.01F × 30V = 0.3C and

€

I =VR

=QRC

=30V

10,000Ω= 0.003A = 3mA .

c. How long does it take for 49% of the initial charge to be dissipated from the capacitor?

€

Q =Qmaxe−

tRC → t = −RC ln Q

Qmax

= −100s× ln

0.51Qmax

Qmax

= 67.3s

Part II: Multiple-Choice Circle your best answer to each question. Any other marks will not be given credit. Each multiple-choice question is worth 3 points for a total of 12 points.

1. Suppose that a capacitor is constructed out of two parallel metal plates with a

dielectric κ inserted between the plates. The capacitor is charged to a maximum value Qmax using a battery of potential difference V and then the capacitor is disconnected from the battery? With the capacitor disconnected from the battery, the dielectric material is removed from the capacitor. Which of the following statements is (are) correct?

a. Qmax remains constant and the electric field between the plates decreases. b. Qmax decreases and the electric field between the plates increases. c. Vmax increases and the electric field between the plates increases d. Vmax increases and the energy stored in the system increases. I. Choice a. only. II. Choice c. only. III. Choices b. and d. IV. Choices c. and d.

2. Suppose that a one-dimensional region of space exists and that over the entire region

there is a constant electric potential. The magnitude of the electric field vector in this region of space

a. must have a constant value at all points in that region. b. must be zero at all points in that region c. cannot be found unless we know its value at a minimum of two points in the

region. d. changes from a large value to a small value as you move throughout the region.

3. Suppose that a charge +Q is fixed at a given location and that a test charge q0 is able

to move. The potential energy of q0 is

a. constant. b. increasing with increasing distance between Q and q0. c. decreasing with increasing distance between Q and q0. d. depends on the acceleration due of q0.

4. An electron is in a circular orbit of radius r around a proton in a hydrogen atom? The

total energy of the electron in the circular orbit is

a.

€

Etotal =ker2

. b.

€

Etotal =ke2

r. c.

€

Etotal =ke2

r2. d.

€

Etotal = −ke2

r

Choice D should have a factor of 2 in the denominator. Since it didn’t choice B or D was accepted.

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