Science 1 Physics Midterm - 23 Feb 1999
SOLUTIONS - see Footnotes!

1.

``Quickie'' Questions   [40 marks - 10 each]

(a)

[10 marks]   On the drawing, sketch in the electric field vectors at points A and B.

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At which point (if either) is the magnitude of E larger? ¹

If you take the electrostatic potential to be zero infinitely far from the charges, is the potential at point A positive, negative or zero? ²

What about the potential at point B? ³

(b)

[10 marks]   A capacitor is made from two metal plates of area A separated by a distance d. The gap is filled with two dielectric slabs of equal thickness but with different dielectric constants $\epsilon_1$ and $\epsilon_2$.

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Calculate the capacitance in terms of the parameters given. ⁴

(c)

[10 marks]   Six identical capacitors of capacitance C=1 F and six identical resistors of resistance $R=1 \; \Omega$ are arranged in a cube as shown.

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If the circuit has a net charge $Q_\circ$ on the capacitors before the switch is closed at time t=0, when will the net charge be $Q_\circ/e$? ⁵

(d)

[10 marks]   The STIRLING CYCLE can be realized using a porous sponge (providing only thermal inertia, no mechanical work) and a flexible diaphragm that changes the volume of an ideal gas in a cylindrical chamber, viewed in cross section below. Identify the steps of the cycle on the p-V diagram, indicating the direction of each step with an arrow on the diagram. ⁶

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Is the system operating as an engine or as a refrigerator? ⁷

2.

Capacitance of the Earth   [30 marks]

Treating the Earth as a spherical conductor and measuring its voltage relative to zero potential at infinite distance, calculate the capacitance C_E of the Earth. ⁸

3.

Cylindrical Shell of Charge   [30 marks] A long cylinder of radius b is made from an insulating material with a uniform charge per unit volume of $\rho$. A cylindrical hole of radius a < b is then bored down its axis. Find the electric field everywhere, assuming that the ends are very far away (compared with b). ⁹