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School of Physics and Astronomy

DEGREE OF BSc & MSci WITH HONOURS

FIRST YEAR EXAMINATION

03 19750

Electromagnetism 1/Temperature & Matter

The total time allowed is 3 hours

MAY/JUNE 2009

In Section 1 candidates may attempt as many questions as they wish. Section 1 contributes 40% of the total mark. Full marks may be achieved by

correct answers to eight questions

In Section 2 candidates must answer all three questions. Note that each question has two parts, of which only one part should be answered. If more than one part is attempted, only the first part will be

marked. Section 2 contributes 60% of the total mark.

Calculators may be used in this examination but must not be used to store text. Calculators with the ability to store text should have their memories deleted

prior to the start of the examination.

PLEASE USE A SEPARATE ANSWER BOOK FOR SECTION 1 AND FOR SECTION 2 QUESTIONS.

A table of physical constants and units that may be required will be found at the end of this question paper.

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SECTION 1

In Section 1 candidates may attempt as many questions as they wish. Marks for all questions attempted will be aggregated up to the maximum; full

marks may be obtained by correct answers to eight questions. This section contributes 40% of the total mark.

Candidates are advised to spend no more than 72 minutes on this section.

Start each Section 1 question on a SEPARATE PAGE of the answer book.

1. The average distance between the proton and the electron in a

hydrogen atom is 5.3 x 10-10 m. Calculate the electrostatic force

acting on the electron, and the electrostatic energy of the system.

Estimate the amount of work needed to completely separate the

electron from the proton. [5]

2. The electric field due to a charged infinite sheet is , where σ

is the surface charge density. If two sheets with identical charge are

placed parallel to each other with a distance d between them, what

is the electric field in between the sheets?

If we put a negative charge Q in between the sheets, will it move if

only electrostatic force is to be considered? [5]

3. Use Gauss’s law to derive expressions for the electric field, both

inside and outside, a uniformly charged sphere of radius R and with

a volume charge density ρ. [5]

4. A long cylindrical conductor of radius R carries a current I which is

uniformly distributed over the cross sectional area of the conductor.

Use Ampere’s law to find the magnetic field due to this current and

sketch how the strength of the magnetic field varies as a function of

distance from the centre of the cylinder. [5]

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5. The figure below shows a circular loop of radius R in a magnetic

field. If the magnetic field changes with time according to

,

and the field lines are perpendicular to the plane of the loop, write

down an expression to show how the magnetic flux passing through

this loop changes with time and hence find the induced emf in the

loop.

If we keep everything the same except that we squeeze the circular

loop into an oval shape, will the induced emf in the oval loop be

greater or smaller than that in the circular loop?

[5] 6. Each circuit shown below consists of three identical resistors of

1Ω resistance. What is the equivalent resistance for each circuit?

(a) (b) (c) (d) [5]

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7. In the circuit shown below, ε=1.2 kV, C=6.5 µF, R1=R2=R3=0.73 MΩ.

With C completely uncharged, switch S is suddenly switched on at

t=0. Determine the current through resistor R1 at t=0 and as .

[5]

8. The total energy U present at any instant in an oscillating LC circuit

is given by:

Assume the circuit has zero resistance, and hence show that the

differential equation that describes the oscillations of a

resistanceless LC circuit is:

. [5]

9. If it takes four minutes to boil a kettle of water which was initially at

, how much longer will it take to boil the kettle dry?

(Ignore the heat capacity of the kettle. The specific heat capacity of

water is and the latent heat of vaporisation of

water is ). [5]

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10. The Lennard-Jones 6-12 potential which describes the

interaction of two molecules is given by

where and are constants.

Sketch indicating the positions of and and the regions of

over which the force between the molecules is

i) repulsive

ii) attractive.

What is the physical significance of and ? [5]

11. Sketch the Maxwell-Boltzmann probability distribution of speeds

of molecules of mass in a gas at temperature .

Explain the physical origins of the main features of the curve. [5]

12. Calculate the root mean squared speed of an argon atom at .

(The molar mass of argon is ). [5]

13. Consider a copper wire of circular cross-section of length and

of diameter . The wire is held under tension. Calculate the

change in wire length when:

i the wire is maintained at constant temperature but its tension is

increased by . [3] ii the tension of the wire is held at its initial value but its temperature

is increased by . [2] (For copper, Young’s modulus, and the coefficient

of linear expansion, ).

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SECTION 2

Section 2 contributes 60% of the total mark. Candidates should attempt all three questions from this section. Note that each

question has two parts of which only one should be answered.

Students should spend no more than 36 minutes on each question. 14. EITHER Part A

(a) State the Biot-Savart law for the magnetic field due to a current

element and give a clear explanation of all the symbols involved in

your equation. [4]

(b) Use this law to show that the magnetic Bx-field at a point on the axis

of a flat circular coil radius a, carrying current I is given by:

,

where x is the distance of the point from the centre of the coil, and the

axis is perpendicular to the plane of the coil. [6] (Question continues on the next page)

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14. A

(c) A spherical shell of radius R with a uniform surface charge density σ,

is rotating about a diameter with angular velocity ω. Show that the

small ring of charge perpendicular to the axis illustrated in the figure

below, represents a current element di given by:

.

By using the result from part (b) above show that the magnetic field at

the centre of the sphere resulting from this current element is given

by:

Hence from this expression determine the total magnetic B-field at the

centre of the sphere.

[You may assume the following definite integral: [10]

ω θ

R

x

dθ

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14. OR Part B

(a) Briefly explain the term electric potential and state the equation how

the electric field can be calculated from the electric potential. [4]

(b) A ring-shaped conductor, of radius a, carries a total charge Q

uniformly distributed around it as illustrated below:

Find the potential at a point P on the ring axis at a distance x from the

centre of the ring. (Hint: consider the electric potential at P due to a

small segment (effectively a point charge) carrying a charge dQ.)

From your expression show that the electric field at P is given by:

[10]

(c) If an electron is placed at the centre of the ring and is then displaced

a small distance x along the axis (x << a), show that the electron

oscillates with a frequency

,

where me is the mass of the electron and e is its charge. [6]

a x P

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15. EITHER Part A (a) State both Kirchhoff’s loop rule and Kirchhoff’s junction rule. [5]

(b) The elements in the electric circuit shown below have the following

values:

The three batteries are ideal batteries with no internal resistance. Find

the magnitude and direction of the current in each of the three

branches.

[12]

(c) Calculate the power dissipation by R2. If the battery ε2 in the above

circuit has a finite internal resistance, what effect will it have on the

power consumption through R2? [3]

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15. OR Part B

(a) In the circuit shown below, capacitor C1 = 3 µF, is charged to a

potential difference V = 5 V. When switch S is closed, charge flows

from C1 to C2. C2 = 8 µF and is uncharged before the switch is closed.

Find the potential difference across each capacitor when charge flow

stops. S

C1 C2

[6]

(b) We now replace C2 in the circuit with an inductor L = 5 mH. When the

switch is closed, charge flow in the circuit gives rise to a time varying

current i. Using the condition that at any time, the voltage across the

capacitor is the same as that across the inductor, show that the

variation of the charge on the capacitor with time can be described

by:

.

Solve the above equation and show that the current in the circuit

oscillates with an angular frequency

.

Obtain numerical values for both ω and the amplitude of the

oscillating current. [10]

(c) For the circuit discussed in (b), the stored electrical energy is:

,

where q0 is the amplitude of the oscillating charge on the capacitor.

Show that the total stored energy within the circuit is independent of

time. [4]

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16. EITHER Part A

a) The figure below shows a thermodynamic process followed by

of gas from an initial state to a final state .

Assuming the gas is ideal and that its molar heat capacity,

calculate : i the work done on the gas [2]

ii the change in internal energy and in the temperature of the gas [8]

iii The amount of heat transferred to the gas [2]

(Question continues on the next page)

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16. A b) A hydroelectric power station brings water from a very large reservoir

high in the mountains down to the generating plant through pipes

running through the mountain. As sketched below, water flows from

the reservoir through a diameter circular pipe near the base of the

reservoir dam beneath the surface of the reservoir. The water

drops through this pipe before entering the turbine via

nozzle.

Ignoring the effects of viscosity and assuming that the water level in

the reservoir remains constant, i what is the speed of the water flowing into the turbine at point A? [3]

ii What is speed of the water flowing from the dam into the pipe? [2]

iii When the flow of water stops (due to a valve in the pipe being

closed), how large is the change in pressure of the water flowing from

the dam into the pipe at point B?

(The density of water is ). [3]

B

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16. OR Part B

a) State the Equipartition Theorem. [3]

Show that the equipartition of energy leads to the following

predictions: i

for a monatomic gas ; [2]

ii for a solid whose atoms are held in a lattice

where is the molar heat capacity at constant volume and is the

Gas constant. [3] Discuss why this prediction (ii) breaks at low temperatures. [4]

b) An ideal gas is admitted into a cylinder at and at a pressure of

.A piston then rapidly compresses the gas from to

. Assuming that for air the ratio of specific heat capacities

, i what are the final temperature and pressure of the gas? [4]

ii Illustrate the compression on a diagram. [2] iii How much work is done to compress the gas?

(You may assume that the molar heat capacity for air,

). [2]