Physics 122 Midterm - 15 Feb. 1995

1.

RC Circuit   [18 marks] A capacitor (C = 8 mF) is initially charged to half of the full charge Q it would have if had been connected to the battery ( ${\cal E} = 6$ V) for a very long time. The circuit contains a resistance (R = 2 $\Omega$). At t=0 the switch S is closed. How long does it take for the capacitor to reach $7\over8$ of its full charge Q?

2.

Orbit Around a Charged String   [18 marks] A long straight cylindrical ``cosmic string" in interstellar space is charged uniformly with a negative electric charge $-\lambda$ per unit length. The resultant electric field acts on a particle of mass m and positive charge q which is in a circular orbit around the string at a radius r. (We assume the string itself has a still smaller radius. We also ignore any possible effects of the string's gravity.)

(a)

Show that the kinetic energy of the particle is given by ${1\over2} m v^2 = k_E q \lambda$ (where $k_E \equiv {1 \over 4\pi\epsilon_\circ}$), regardless of the value of m or r.

(b)

If the charge on the wire is $\lambda = 1.0$ C/m and the particle has a charge of q = 1.0 C and a mass of 0.2 kg, show that the particle is moving at 0.1% of the speed of light.

(c)

If the radius of the particle's orbit is r = 0.1 m, what is the magnetic field at the centre of its orbit due to the circulating charge?

3.

Athletic Potential   [14 marks] A sprinter on the UBC track team runs at a speed of 8 m/s while holding a 5 m long pole vault pole horizontal at right angles to her direction of motion.¹ The pole has a fine copper wire down the center from one end to the other. If the Earth's magnetic field at the location of the runner has a magnitude of $0.3 \times 10^{-4}$ T and makes an angle of $45^\circ$ with the horizontal, what is the potential difference [in volts] between one end of the pole and the other?

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