PHY250B

1 PHY250B TURN OVER

Data Provided: A formula sheet, equation sheet for electromagnetism and a table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester (2019-2020) PHY250 - From electromagnetism to quantum mechanics Paper B – Electromagnetism and quantum mechanics 3 HOURS

Instructions:

Answer ALL questions.

There are two sections to this paper. Each section has two questions. Answer each

section in a separate book.

All questions are marked out of 25. The breakdown on the right-hand side of the paper

is meant as a guide to the marks that can be obtained from each part.

PHY250B

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SECTION 1 – Electromagnetism

Equation sheets

Electrostatics Coulomb’s law and definition of electric field

1 22

0

ˆ

4

Q Q

r

rF , 0

lim /t

tQ

Q

E F, qF E

Electric field due to a point charge and Gauss’ law for electrostatics

20

ˆ

4

Q

r

rE ,

0

1d

SQ

E S ,

Potential energy of two point charges, definition of electric potential, electric potential of a point charge and E/V relationships

1 2

04

Q QU

r ,

lim /0

BA BA t

t

V U QQ

, 0

( )4

QV r

r , dV V E E L

Definition of capacitance, energy stored by a capacitor, energy density of an electric field

Q=CV, 2

21

2 2

QU CV

C , 2

0

1

2u E

Electric potential and electric fields produced by an electric dipole

V Pp

r( )

r

4 02

Ep

rr

2

4 03

cos

Ep

r

sin

4 03

Torque and potential energy of an electric dipole

T p E , U p E

Magnetism Biot-Savart Law and Ampere’s Law

02

ˆdd

4

I

r

L rB , 0d

LI B L

Force on a current element and moving charged particle dF=IdLB, F=qvB Electromagnetic induction Definitions of magnetic flux and electromotance

dS

B S , dL

V E L

Faradays law, definition of inductance, voltage in an inductor when the current changes, energy stored by an inductor and energy density of a magnetic field

d

dV

t

, =LI,

d

d

IV L

t , 21

2MU LI ,

2

0

1

2M

Bu

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Dielectrics and magnetic materials Relationships between quantities and definitions of susceptibility

D=0rE, D=0E+P, d fSQ D S , P=0eE, r=1+e, H=B/(0r), 0 ( ) B H M ,

d cLI H L , M=mH, r=1+m

Boundary conditions in the absence of free charges and currents Normal components of D and B are continuous. Tangential components of E and H are continuous Electromagnetic waves Maxwell’s Equations

0 0 0

0

0f

r r r

r t t

c

B EE E B B J

Relationship between field magnitudes E=cB Poynting Vector, magnitude of Poynting vector and Radiation pressure for an absorbing surface N E H , N=E2/(c0), P=N/c

Refractive index for a non-magnetic medium n2 = r Power transmission coefficient for normal incidence

1 2

2

1 2

4n

n nT

n n

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1. a) A solid non-conducting sphere of radius a has a total charge +Q distributed

uniformly throughout its volume. A second thin walled hollow sphere of radius b (>a) has a total charge –Q and is placed concentric with the first sphere. Apply Gauss’ law to show that the magnitude of the electric field in the region

0ra is given by:

3

0

( )4

QrE r

a .

Find the magnitude of the electric field in the two regions a<r<b and rb, and sketch the dependence of the electric field on the radial distance r. [6]

b) If the non-conducting sphere is replaced with a conducting one how are the results from part a) modified? Justify your answers. [3]

c) If the region between the spheres is filled with a dielectric of breakdown strength

2.0104 V m-1 and relative permittivity 2.0, what is the maximum possible value of

Q for a value of a = 2.0 cm? [2]

d) An electric dipole of moment 8.010-12 C m is placed in an electric field of

magnitude 2.0105 V m-1. Find the maximum and minimum potential energy of the dipole and sketch the relative orientations of the dipole and field for these two cases. [3]

e) An infinitely long thin straight wire carries a current I. Using Ampere’s law, or otherwise, show that the magnitude of the magnetic field a distance r from the wire is given by:

0( )2

IB r

r

. [3]

f) An electron travels with a velocity 6.0105 m s-1 at an initial distance of 5.0 mm parallel to a wire carrying a current of 5.0 A. The velocity of the electron is in the opposite direction to that of the electrons which produce the current flow in the wire. Using the result from part e) find the initial magnetic force (both magnitude and direction) which acts on the electron. [3]

g) A laser beam has a total power of 10 W and a diameter of 5.0 mm. Calculate the average electric and magnetic fields due to this beam. [3]

h) A small reflecting disk of radius 1.0 mm is placed in the laser beam defined in part g). What is the magnitude of the radiation force acting on this disk? [2]

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2. a) A circular coil of radius r and N turns is placed in a uniform magnetic field B.

The normal to the coil makes an angle to the direction of the field. Show that the magnetic flux through the coil is given by:

2( ) cos( )BN r . [3]

b) If the coil defined in part a) is rotated with a constant angular frequency derive an

equation for the time dependence of the induced voltage. The rotation is such that the direction of the normal to the coil rotates. [3]

c) What is the maximum voltage induced in the system defined in parts a) and b) for values B=0.5 T, N= 100, r= 2.0 cm and frequency of rotation 100 Hz? [3]

d) A rectangular coil of sides length L and w, and N turns, is placed parallel to a long straight wire carrying a current I. The nearest edge of the coil is a distance a from the wire (see figure).

Show that the magnetic flux linking the coil is given by:

0 ln 12

ILN w

a

. [4]

[You may assume the result for the magnitude of the magnetic field a distance r

from a long straight wire carrying a current I: 0( )2

IB r

r

]

e) If the current in the system defined in part d) varies with time as I(t)=t, where is a constant, derive a result for the voltage induced in the coil. [3]

f) State each of the four Maxwell’s equations and briefly explain the physics they describe. Where relevant, state the law(s) that each equation relates to. [7]

g) State the four Maxwell equations for the case of a vacuum. [2]

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SECTION 2 – Quantum Mechanics

3.

a) Write down the one-dimensional time-independent Schrödinger equation, and

define all the terms in the equation. Leave your answer in the most general

form with V(x) for the potential. [2]

Consider a particle of mass m confined in the following one-dimensional potential V(x):

,2

3( ) 0 ,

2 2

3.

2

Lfor x

L LV x for x

Lfor x

b) Write down the wavefunctions for a particle in the infinite square well potential

described above. Hint: recall from lectures that the wave functions for a particle in

an infinite square well with walls at x = 0 and x = L are:

2( ) sinn

n xx

L L

. [4]

c) Write down and sketch the wavefunctions for each of the n = 1, n = 2 and n = 3

states. Sketch each of these functions and correctly label your axes. [4]

For parts d), e) and f) use wavefunctions of the form:

2( ) sinn

n xx

L L

which correspond to a particle in an infinite depth well extending from x=0 to x=L.

d) Show by explicit calculation, that the probability that the particle is confined to the

entire width of the well is one. [4]

e) Determine the probability that the particle is confined to the first 1/a of

the width of the well. [4]

f) Comment on the n-dependence of the probability found in part e). [2]

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The nuclear potential that binds protons and neutrons in the atomic nucleus can often be

approximated by an infinite square well. Assume that a proton (mass 938 MeV / c2) is

confined to an infinite square well of width 10 fm.

g) Write down an expression for the energy, En, of a particle of mass m in the nth

energy state of an infinite square well with a width L. Hint: Consider the allowed k-

values and the relationship between k and E. [2]

h) Calculate the energy and wavelength of the photon emitted when the proton

undergoes a transition from the first excited state (n = 2) to the ground state (n = 1). [3]

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4.

a) Consider a wavefunction of the form

( )x

x Ae

where A and are constants. Calculate the wavefunction in momentum space, (p). [4]

b) Calculate the normalisation constant, A, for the wavefunction defined in part a). [3]

c) Operators ˆ ˆˆ, andA B C ,which do not commute with one another, obey a number of

relationships. Prove the following:

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ[ ,[ , ]] [ ,[ , ]] [ ,[ , ]] 0A B C B C A C A B [4]

In the remaining parts of this question, a particle is in a state described by the wavefunction

2

1/42

( ) axax e

where a is a constant and −∞ < x < ∞.

d) Calculate < x >, < x 2 > and ∆x. [6]

e) Calculate < p >, < p2 > and ∆p. [6]

f) Verify that the product ∆p∆x is consistent with the predictions from the uncertainty

principle. Hint: Recall that the uncertainty in a general quantity A is given by

22A A A . [2]

END OF EXAMINATION PAPER

PHYSICAL CONSTANTS &MATHEMATICAL FORMULAE

Physical Constants

electron charge e = 1.60×10−19 Celectron mass me = 9.11×10−31 kg = 0.511MeV c−2proton mass mp = 1.673×10−27 kg = 938.3MeV c−2neutron mass mn = 1.675×10−27 kg = 939.6MeV c−2Planck’s constant h = 6.63×10−34 J sDirac’s constant (~ = h/2π) ~ = 1.05×10−34 J sBoltzmann’s constant kB = 1.38×10−23 J K−1 = 8.62×10−5 eVK−1speed of light in free space c = 299 792 458 ms−1 ≈ 3.00×108 ms−1permittivity of free space ε0 = 8.85×10−12 Fm−1permeability of free space µ0 = 4π×10−7 Hm−1Avogadro’s constant NA = 6.02×1023 mol−1gas constant R = 8.314 Jmol−1K−1ideal gas volume (STP) V0 = 22.4 l mol−1gravitational constant G = 6.67×10−11 Nm2 kg−2Rydberg constant R∞ = 1.10×107 m−1Rydberg energy of hydrogen RH = 13.6 eVBohr radius a0 = 0.529×10−10 mBohr magneton µB = 9.27×10−24 J T−1fine structure constant α ≈ 1/137Wien displacement law constant b = 2.898×10−3 mKStefan’s constant σ = 5.67×10−8 Wm−2K−4radiation density constant a = 7.55×10−16 Jm−3 K−4mass of the Sun M� = 1.99×1030 kgradius of the Sun R� = 6.96×108 mluminosity of the Sun L� = 3.85×1026 Wmass of the Earth M⊕ = 6.0×1024 kgradius of the Earth R⊕ = 6.4×106 m

Conversion Factors1 u (atomic mass unit) = 1.66×10−27 kg = 931.5MeV c−2 1 Å (angstrom) = 10−10 m1 astronomical unit = 1.50×1011 m 1 g (gravity) = 9.81 ms−21 eV = 1.60×10−19 J 1 parsec = 3.08×1016 m1 atmosphere = 1.01×105 Pa 1 year = 3.16×107 s

Polar Coordinates

x = r cos θ y = r sin θ dA = r dr dθ

∇2 =1

r

∂

∂r

(r∂

∂r

)+

1

r2∂2

∂θ2

Spherical Coordinates

x = r sin θ cosφ y = r sin θ sinφ z = r cos θ dV = r2 sin θ dr dθ dφ

∇2 =1

r2∂

∂r

(r2∂

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1

r2 sin2 θ

∂2

∂φ2

Calculusf(x) f ′(x) f(x) f ′(x)

xn nxn−1 tanx sec2 x

ex ex sin−1(xa

)1√

a2−x2

lnx = loge x1x

cos−1(xa

)− 1√

a2−x2

sinx cosx tan−1(xa

)a

a2+x2

cosx − sinx sinh−1(xa

)1√

x2+a2

coshx sinhx cosh−1(xa

)1√

x2−a2

sinhx coshx tanh−1(xa

)a

a2−x2

cosecx −cosecx cotx uv u′v + uv′

secx secx tanx u/v u′v−uv′v2

Definite Integrals∫ ∞0

xne−ax dx =n!

an+1(n ≥ 0 and a > 0)

∫ +∞

−∞e−ax

2 dx =

√π

a∫ +∞

−∞x2e−ax

2 dx =1

2

√π

a3

Integration by Parts:∫ b

a

u(x)dv(x)dx

dx = u(x)v(x)∣∣∣ba−∫ b

a

du(x)dx

v(x) dx

Series Expansions

Taylor series: f(x) = f(a) +(x− a)

1!f ′(a) +

(x− a)2

2!f ′′(a) +

(x− a)3

3!f ′′′(a) + · · ·

Binomial expansion: (x+ y)n =n∑k=0

(n

k

)xn−kyk and

(n

k

)=

n!

(n− k)!k!

(1 + x)n = 1 + nx+n(n− 1)

2!x2 + · · · (|x| < 1)

ex = 1+x+x2

2!+x3

3!+ · · · , sinx = x− x

3

3!+x5

5!−· · · and cosx = 1− x

2

2!+x4

4!−· · ·

ln(1 + x) = loge(1 + x) = x− x2

2+x3

3− · · · (|x| < 1)

Geometric series:n∑k=0

rk =1− rn+1

1− r

Stirling’s formula: logeN ! = N logeN −N or lnN ! = N lnN −N

Trigonometry

sin(a± b) = sin a cos b± cos a sin b

cos(a± b) = cos a cos b∓ sin a sin b

tan(a± b) = tan a± tan b

1∓ tan a tan b

sin 2a = 2 sin a cos a

cos 2a = cos2 a− sin2 a = 2 cos2 a− 1 = 1− 2 sin2 a

sin a+ sin b = 2 sin 12(a+ b) cos 1

2(a− b)

sin a− sin b = 2 cos 12(a+ b) sin 1

2(a− b)

cos a+ cos b = 2 cos 12(a+ b) cos 1

2(a− b)

cos a− cos b = −2 sin 12(a+ b) sin 1

2(a− b)

eiθ = cos θ + i sin θ

cos θ =1

2

(eiθ + e−iθ

)and sin θ =

1

2i(eiθ − e−iθ

)cosh θ =

1

2

(eθ + e−θ

)and sinh θ =

1

2

(eθ − e−θ

)Spherical geometry:

sin a

sinA=

sin b

sinB=

sin c

sinCand cos a = cos b cos c+sin b sin c cosA

Vector Calculus

A ·B = AxBx + AyBy + AzBz = AjBj

A×B = (AyBz − AzBy) i+ (AzBx − AxBz) j+ (AxBy − AyBx) k = εijkAjBk

A×(B×C) = (A ·C)B− (A ·B)C

A · (B×C) = B · (C×A) = C · (A×B)

gradφ = ∇φ = ∂jφ =∂φ

∂xi+

∂φ

∂yj+

∂φ

∂zk

divA = ∇ ·A = ∂jAj =∂Ax∂x

+∂Ay∂y

+∂Az∂z

curlA = ∇×A = εijk∂jAk =

(∂Az∂y− ∂Ay

∂z

)i+

(∂Ax∂z− ∂Az

∂x

)j+

(∂Ay∂x− ∂Ax

∂y

)k

∇ · ∇φ = ∇2φ =∂2φ

∂x2+∂2φ

∂y2+∂2φ

∂z2

∇×(∇φ) = 0 and ∇ · (∇×A) = 0

∇×(∇×A) = ∇(∇ ·A)−∇2A