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UNIVERSITY OF LONDON

BSc EXAMINATION 1999

For Internal Students ofRoyal Holloway

A*

DO NOT TURN OVER UNTIL TOLD TO BEGIN. .

PH2420B: ELECTROMAGNETISMPH2420A(R): ELECTROMAGNETISM - PAPER FOR RESIT CANDIDATESTime Allowed: TWO hours

Answer QUESTION ON E and TWO other questionsNo credit will be given for attempting anyfirther questions

Approximate pad-marksfor questions are given in the right-hand margin

Calculators ARE permittedPHZ420B and PH2420A(R)/50ORoyal Holloway and Bedford New College 1999

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GENERAL PHYSICAL CONSTANTSPermeability of vacuumPermittivity of vacuum

Speed of light in vacuumElementary chargeElectron (rest) massUnified atomic mass constant

4-Proton rest massNeutron rest massRatio of electronic charge tomassPlanck constant

Boltzmann constantStefan-Boltzmann constantGas constantAvogadro constantGravitational constantAcceleration due to gravityVolume of one mole of an ideal gas at STPOne standard atmosphere

H m*'F m-'mF'm i1Ckgkgkgksc kg1J SJ SJ K-'W me2K~J mol" K"mol-'N m2 kg'2m s-2m3N m-2

MATHEM ATICAL CONSTANTSe = 2.718 a = 3.142 lo&lO= 2.303

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- page 1- PHZ42OB and PHZ42OA(R)PART

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ANSWEROWL Y FIVE sections of Question OneYou are advi:.t:d not to spend more than 40 minutes answering Qu do n One

1. (a) Calculate the ratio of the electrostatic force to the gravitational force between apro~onnd an electron. How does this ratio depend ontheparticle separation? 141

(b) Describe and explain qualitatively a possible mechanism for the finite electricalconductivity of metals. 141

(c) Explain the meaning of the.statement that an electric field is conservative. When isan electric field not conservative?L (41

(d) Assuming the Poisson equation

show that the electric field in a cavity within a conductor is zero.(e) How is the Ampere defined? Describe how, following from this definition, electric

currents can be measured M(f) Show how an arrangement of electric and magnetic fields can be used to select a

beam of charged particles moving with a given velocity. 141

TURN OVER

a

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- page 2 - PH242OB and PH2420A(R)PA' -1

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2. (a) State Gauss's law relating the fluxof an electric field E out of a closed surfiice to thetotal electric charge enclosed. 141

(b) A capacitor consists of two parallel conducting plates each of area A separated by adistance d in fiee space. One plate carries a charge of +Q and the other -Q. Byusing Gauss's law, or otherwise, show that the electric field between the plates hasmagnitude . . I . -,

0.c . $ , C # F -9-Q ~ t l ' i r *! E= -*- &,,A ' v

(c) The above result implies that the electric field between the plates is unifonn. Explaincarefully the arguments leading to the uniformity of the field, including a discussionof the limitation of such arguments. r31

(d) Define the electric potential V. Show that the potential difference between the platesis given by

(e) Hence demonstrate that the potential difference is proportional to the charge on theplates and show that the capacitanceC is given by

( f ) Outline briefly the effect of placing a dielectric material between the plates.

TURNOVER

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- page 3 - PH2420B and PH2420A(R)PART

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3. (a) Charges +g and -q are separated by a d i c e x. State the magnitude and thedirection of the resultant dipole moment p. 131

(b) Show that a dipole experiences no force in a uniform electric field. I31(C) Show that a dipole moment p experiences a torque given by p X E in a uniform

electric field E.(d) Show that the electric potential at the point P in the figure is givenby

(e) Show that for r >>X the electric potential at P may be approximated by

(f) Explain why a dipole experiences no force in a urtifonn electric field, but it doesexperience a force in a non-uniform ield.

TURNOVER-- -- ~ -- -~-- --

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4. (a) Show that Ampere's law relating the line integral of the magnetic field B around aclosed loop to the current I passing through the loop,

may be expressed in terms of the curl of B ascurlB=p,J .

(b) Demonstrate that there is an inconsistency in Ampere's law 'when applied to the [S]space between the plates of gmpacitor when it is charging up.

(c) Since the charge on a capacitor in vacuo may be written as

where E is the electric field and A is the area of the plates, show that theinconsistency may be removed through the introduction of a fictitious current igiven by

(d) Show that, including the effect of this current, the curl of B may be written as

(e) In terms of the electric f lux comment on the parallels with Faraday's law ofelectromagnetic induction

5 . (a) Describe an experiment to verifj. he inverse square law of the Coulomb force to thehighest possible accuracy. You should give particular attention to the methodswhereby this high accuracy is obtained and errors minimised. (171

(b) Discuss a possible explanation ifit were to be discovered that the inverse square lawwere not quite correct. 131

END