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Physics 321 – Electromagnetism I First midterm 9.30 am Wednesday 31 October Autumn 2013 Instructor: David Cobden A118

Do not turn this page or write anything until I say go, which will be at 9.30.

Hand your exam to me before I leave the room at 10.55.

Attempt all the questions.

Please write your name on every page and your SID on the first page.

Write all your working on these question sheets. Use this front page for extra working. It is important to show your calculation or derivation. Some marks are given for showing clear and accurate working and reasoning.

Watch the blackboard for corrections and clarifications during the exam.

This is a closed book exam. No books, notes, or calculators allowed.

1. [16] Write carefully the 6 integral and differential equations relating each pair of fields arranged in the triangle below.

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2. [8] If the electric field is everywhere, find the total charge enclosed in a sphere of radius . 3. [5] If you draw the electric field of a point charge on a piece of paper as straight radial lines like the spokes of a wheel, does the density (inverse spacing) of the lines accurately reflect the decay of the electric field with distance? Why or why not? 4. [6] State the uniqueness theorem for Laplace’s equation, mentioning both types of boundary conditions that are sufficient. 5. [8] A point charge and another are located on opposite sides of a neutral conducting sphere as indicated below. Sketch the lines of electric field and equipotentials and indicate any induced charges.

6. [6] A charge is now added to the sphere. Again sketch the field lines and equipotentials.

+q -q

+q +q -q

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7. [5] A cross-section through a smooth egg-shaped conductor is shown below. It holds charge and is at potential . Define the self-capacitance and state the relationship between and .

8. [4] There is an ambiguous quantity here. What is it? 9. [4] If is sufficiently negative electrons will start to be field-emitted from the surface (perhaps making a spark). Why are they not emitted when is small? 10. [4] Where on the egg’s surface will they be emitted first, and why? (Sketch field lines if needed) 11. [12] A spherical capacitor (rather weird, admittedly) consists of two concentric spherical conducting shells of radii and ( ) with vacuum in between and outside them. It holds a charge , ie, the outer has positive charge and the inner . Find the mutual capacitance between them (by finding the potential difference).

12. [6] Find the electrostatic pressure acting on the outer conductor (including its direction). 13. [8] Find the energy stored by integrating the energy density in the electric field. 14. [8] Find the same answer for the energy stored by a different means.

,

a b

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15. [12] Two particles, one of mass and the other of mass , both having charge , are initially fixed at a separation . They are then released. What are their speeds, and , when their separation reaches 2 ? 16. [20] A circular ring of radius carries a uniform line charge density . It is fixed with its axis vertical at a distance above a infinite horizontal grounded conducting surface. Find the electrostatic potential and the electric field along the axis of the ring (above the surface only).