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95.144 Final Exam Fall 2015

Section instructor____________________ Section number__________ Last/First name__________________________________________________________ Last 3 Digits of Student ID Number: __________ Show all work. Show all formulas used for each problem prior to substitution of numbers. Label diagrams and include appropriate units for your answers. You may use an alphanumeric calculator during the exam as long as you do not program any formulas into memory. By using an alphanumeric calculator you agree to allow us to check its memory during the exam. Simple scientific calculators are always OK!

A Formula Sheet Is Attached To The Back Of This Examination Be Prepared to Show your Student ID Card

Score on each problem:

1. (30) ____ 2. (20) ____ 3. (20) ____ 4. (20) ____ 5. (20) ____ 6. (20) ____ 7. (20) ____

Total Score (out of 150 pts) ____

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1. Conceptual Questions (30 point)

1.1. (5pts) A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if a piece of dark paper is lowered to cover the top half of the lens? A. An inverted but blurry image.

B. An image that is dimmer but

otherwise unchanged.

C. Only the top half of the image.

D. Only the bottom half of the image.

E. No image at all.

1.2. (5pts) Four point charges are arranged at the corners of a square. Find the electric field E and the potential V at the center of the square.

A) E=0; V=0 B) E=0; V≠0 C) E≠0; V≠0 D) E≠0; V=0 E) E = V regardless of the value

1.3. (5pts) A parallel-plate capacitor initially has a potential difference of 400 V and is then disconnected from the charging battery. If the plate spacing is now doubled, what is the new value of the voltage?

A) 100 V B) 200 V C) 400 V D) 800 V E) 1600 V

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1.4 (5pts) Figure is a snapshot graph of two plane waves passing through a region of space. Each wave has a 2.0 mm amplitude and the same wavelength. What is the net displacement of the medium at points A and B?

1.5 (10pts) The figure shows six conceivable trajectories of light rays leaving an object. Which, if any, of these trajectories are impossible? For each that is possible, what are the requirements of the index of refraction n2? (2pts) Impossible _________ (2pts) Requires n2> n1 _________ (2pts) Requires n2= n1 _________ (2pts) Requires n2< n1 _________ (2pts) Possible for any n2 _________

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Problem 2. (20 pts) A 4.0-cm-tall cat is placed 20 cm from a concave mirror. The mirror’s radius of curvature is 100 cm.

a) Determine the image position using ray tracing (draw it). Identify if the image is upright/inverted and real/virtual.

b) Calculate the image position and height.

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Problem 3. (20 pts) The figure shows a laser beam deflected by a 30°-60°-90° prism. What is the prism’s index of refraction?

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Problem 4. (20 pts) A double slit is illuminated simultaneously with orange light of λor=636 nm and light of an unknown wavelength. The m=6 bright fringe of an unknown wavelength overlaps the m=5 bright orange fringe. What is the frequency of the unknown light?

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Problem 5. (20 pts) A plane electromagnetic wave propagates in a vacuum in the +z direction. The wave has a wavelength of 10 m and the electric field is along the x direction and has an amplitude of is 0.30 V/m, with one maximum at z = 0 and t = 0.

a) What is the wavenumber?

b) What is the angular frequency?

c) What is the mathematical expression for the magnetic field?

d) What is the mathematical expression for the electric field?

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Problem 6. (20 pts) A proton whose mass is 1.67x10-27 kg is accelerated from rest by a voltage 3000 V. Then, it enters an area with a uniform magnetic field (0.34 T) which is perpendicular to the direction of proton’s motion (see figure).

a) What is the proton’s speed when it enters the magnetic field? b) What is the radius of curvature as the proton moves through the magnetic

field? c) What is its period of revolution?

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Problem 7. (20 pts) A 2.5-cm-radius coil has 20 turns and a resistance of 0.5 Ω. A magnetic field perpendicular to the coil is 𝑩𝑩 = 𝟎𝟎.𝟎𝟎𝟎𝟎𝟎𝟎 + 𝟎𝟎.𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎, where B is in tesla and t is in seconds.

a) Find an expression for the induced current I(t) as a function of time. b) Evaluate I at t = 5 s and t = 10 s?

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Formula Sheet: Electricity and Magnetism Coulomb’s law

𝐹𝐹 = 𝑘𝑘𝑞𝑞𝑞𝑞𝑟𝑟2

Electric Field

𝐸𝐸�⃗ =�⃗�𝐹𝑞𝑞

Field of a point charge

𝐸𝐸 = 𝑘𝑘𝑞𝑞𝑟𝑟2

Electric field inside a capacitor

𝐸𝐸 =𝜂𝜂𝜀𝜀0

Principle of superposition

𝐸𝐸�⃗ 𝑛𝑛𝑛𝑛𝑛𝑛 = �𝐸𝐸�⃗ 𝑖𝑖

𝑁𝑁

𝑖𝑖=1

Electric flux

Φ𝐸𝐸 = �𝐸𝐸�⃗ ∙ 𝑑𝑑𝐴𝐴

Gauss’s law

Φ = �𝐸𝐸�⃗ ∙ 𝑑𝑑𝐴𝐴 =𝑞𝑞𝑖𝑖𝑛𝑛𝜀𝜀0

Electric potential

𝑉𝑉 =𝑈𝑈𝑞𝑞

ΔV = 𝑉𝑉𝑓𝑓 − 𝑉𝑉𝑖𝑖 = −�𝐸𝐸�⃗ ∙ 𝑑𝑑𝑠𝑠

𝑓𝑓

𝑖𝑖

For a point charge 𝑉𝑉(𝑟𝑟) = 14𝜋𝜋𝜀𝜀0

𝑄𝑄𝑟𝑟

For a paralle-plate capacitor 𝑉𝑉 = 𝐸𝐸𝑠𝑠

Potential Energy 𝑈𝑈 = 𝑞𝑞𝑉𝑉

Two point charges

𝑈𝑈 = 𝑘𝑘𝑞𝑞𝑞𝑞𝑟𝑟

Capacitors

𝐶𝐶 =𝑞𝑞Δ𝑉𝑉

Parallel-plate 𝐶𝐶 = 𝜀𝜀0𝐴𝐴𝑑𝑑

Capacitors connected in parallel 𝐶𝐶𝑛𝑛𝑒𝑒 = 𝐶𝐶1 + 𝐶𝐶2 + ⋯ Capacitors connected in series

1𝐶𝐶𝑛𝑛𝑒𝑒

=1𝐶𝐶1

+1𝐶𝐶2

+ ⋯

Energy stored in a capacitor 𝑈𝑈 = 𝑄𝑄2

2𝐶𝐶

Ohm’s law 𝑉𝑉 = 𝐼𝐼𝐼𝐼

𝐼𝐼 =𝑑𝑑𝑞𝑞𝑑𝑑𝑑𝑑

𝐼𝐼 = 𝜌𝜌𝑙𝑙𝐴𝐴

�𝐼𝐼𝑖𝑖𝑛𝑛 = �𝐼𝐼𝑜𝑜𝑜𝑜𝑛𝑛

�Δ𝑉𝑉𝑖𝑖 = 0

Power 𝑃𝑃 = 𝐼𝐼𝑉𝑉

Resistors connected in series 𝐼𝐼𝑛𝑛𝑒𝑒 = 𝐼𝐼1 + 𝐼𝐼2 + 𝐼𝐼3 + ⋯

Resistors connected in parallel 1𝐼𝐼𝑛𝑛𝑒𝑒

=1𝐼𝐼1

+1𝐼𝐼2

+1𝐼𝐼3

+ ⋯

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The potential difference across a charging capacitor in RC circuit

𝑉𝑉(𝑑𝑑) = 𝜀𝜀(1 − 𝑒𝑒−𝑛𝑛 𝑅𝑅𝐶𝐶� ) A magnetic field exerts a force

𝑑𝑑𝐹𝐹�����⃗ = 𝐼𝐼𝑑𝑑𝑙𝑙���⃗ × 𝐵𝐵�⃗ �⃗�𝐹 = 𝐼𝐼𝑙𝑙 × 𝐵𝐵�⃗ �⃗�𝐹 = 𝑞𝑞�⃗�𝑣 × 𝐵𝐵�⃗

The Biot-Savart Law

𝐵𝐵�⃗ =𝜇𝜇0𝑞𝑞�⃗�𝑣 × �̂�𝑟

4𝜋𝜋𝑟𝑟2

𝑑𝑑𝐵𝐵�⃗ =𝜇𝜇0𝐼𝐼𝑑𝑑𝑠𝑠 × �̂�𝑟

4𝜋𝜋𝑟𝑟2

The magnetic field of: A straight line wire

𝐵𝐵 =𝜇𝜇0𝐼𝐼2𝜋𝜋𝑟𝑟

A solenoid 𝐵𝐵 = 𝜇𝜇0𝑛𝑛𝐼𝐼

Magnetic flux

Φ𝐵𝐵 = �𝐵𝐵�⃗ ∙ 𝑑𝑑𝐴𝐴

Inductance

𝐿𝐿 =Φ𝐵𝐵

𝐼𝐼

𝐿𝐿 =µ0𝑁𝑁2𝐴𝐴

𝑙𝑙

𝜀𝜀 = −𝐿𝐿𝑑𝑑𝐼𝐼𝑑𝑑𝑑𝑑

Energy stored in an inductor

𝑈𝑈 = 𝐿𝐿 𝐼𝐼2

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“Discharged” LR circuit 𝐼𝐼 = 𝐼𝐼0𝑒𝑒−

𝑛𝑛 𝜏𝜏� ; 𝜏𝜏 = 𝐿𝐿/𝐼𝐼

Maxwell’s equations

�𝐸𝐸�⃗ ∙ 𝑑𝑑𝐴𝐴�����⃗ =𝑞𝑞𝜀𝜀0

�𝐵𝐵�⃗ ∙ 𝑑𝑑𝐴𝐴�����⃗ = 0

Ɛ = �𝐸𝐸�⃗ ∙ 𝑑𝑑𝑠𝑠����⃗ = −𝑑𝑑Φ𝐵𝐵

𝑑𝑑𝑑𝑑

�𝐵𝐵�⃗ ∙ 𝑑𝑑𝑠𝑠����⃗ = 𝜇𝜇0𝐼𝐼 + 𝜇𝜇0𝜀𝜀0𝑑𝑑Φ𝐸𝐸

𝑑𝑑𝑑𝑑

�⃗�𝐹 = 𝑞𝑞(𝐸𝐸�⃗ + �⃗�𝑣 × 𝐵𝐵�⃗ )

The Poynting vector

𝑆𝑆 =1𝜇𝜇0

(𝐸𝐸�⃗ × 𝐵𝐵�⃗ )

𝐸𝐸0 = 𝑐𝑐𝐵𝐵0

Malus’s Law

𝐼𝐼 = 𝐼𝐼𝑐𝑐𝑐𝑐𝑠𝑠2𝜃𝜃

Traveling Wave

𝑦𝑦(𝑥𝑥, 𝑑𝑑) = 𝐴𝐴𝑠𝑠𝐴𝐴𝑛𝑛(𝑘𝑘𝑥𝑥 − 𝜔𝜔𝑑𝑑 + 𝜑𝜑0)

𝑘𝑘 =2𝜋𝜋𝜆𝜆 ; 𝜔𝜔 =

2𝜋𝜋𝑇𝑇 ; 𝑣𝑣 = 𝜆𝜆𝜆𝜆; 𝑣𝑣 =

𝜔𝜔𝑘𝑘

Interference

Δ𝜑𝜑 = 2𝜋𝜋Δ𝑟𝑟𝜆𝜆 + Δ𝜑𝜑0 = 𝑚𝑚2𝜋𝜋 (𝑐𝑐𝑐𝑐𝑛𝑛𝑠𝑠𝑑𝑑𝑟𝑟)

Δ𝜑𝜑 = 2𝜋𝜋Δ𝑟𝑟𝜆𝜆 + Δ𝜑𝜑0

= (𝑚𝑚 +12)2𝜋𝜋 (𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑟𝑟)

𝐴𝐴 = �2acos (Δ𝜑𝜑2 )�

Standing Waves

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A(x)=2aSin(kx)

𝜆𝜆𝑚𝑚 =2𝐿𝐿𝑚𝑚 ; 𝜆𝜆𝑚𝑚 = 𝑚𝑚

𝑣𝑣2𝐿𝐿

Double Slit

𝑦𝑦𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑑𝑑

Diffraction grating

𝑑𝑑 sin𝜃𝜃𝑚𝑚 = 𝑚𝑚𝜆𝜆

𝑦𝑦𝑚𝑚 = 𝐿𝐿 tan𝜃𝜃𝑚𝑚

Thin-lens equation: 1𝜆𝜆 =

1𝑠𝑠 +

1𝑠𝑠′

𝑚𝑚 = −𝑠𝑠′

𝑠𝑠 ; |𝑚𝑚| =ℎ′

ℎ

Snell’s Law:

𝑛𝑛1 sin𝜃𝜃1 =𝑛𝑛2 sin𝜃𝜃2

TIR: sin𝜃𝜃𝑐𝑐 =𝑛𝑛2𝑛𝑛1

Constants

Charge on electron 𝑒𝑒 = 1.60 ∙ 10−19 𝐶𝐶 Electron mass 𝑚𝑚 = 9.11 ∙ 10−31 𝑘𝑘𝑘𝑘 Proton mass 𝑚𝑚 = 1.67 ∙ 10−27 𝑘𝑘𝑘𝑘 Permittivity of free space

𝜀𝜀0 = 8.85 ∙ 10−12 𝐶𝐶2/𝑁𝑁𝑚𝑚2 Permeability of free space

𝜇𝜇0 = 4𝜋𝜋 ∙ 10−7 𝑇𝑇𝑚𝑚/𝐴𝐴

𝑘𝑘 =1

4𝜋𝜋𝜀𝜀0= 8.99 ∙ 109 𝑁𝑁𝑚𝑚2/𝐶𝐶2

𝑐𝑐 = 1�𝜀𝜀0𝜇𝜇0

= 3.0 ∙ 108𝑚𝑚/𝑠𝑠

Kinematic eq-ns with const. Acc.: v(t) = v0x+at x(t) = x0+ v0xt +(1/2) at2 v2 = v0x

2 + 2a(x – x0) Centripetal acceleration 𝑎𝑎𝑅𝑅 = 𝑣𝑣2 𝑟𝑟⁄ L=2πR A=πR2

V=(4/3)πR3