PHYSICS 241E - Test No. 2
October 28, 1996

1) An infinitely long conducting cylinder of radius A = 1 cm with a linear charge density of = 3 nC/m is centred on the axis. It is surrounded by a conducting cylinder having an inner radius B = 3 cm and outer radius C = 4 cm carrying a net charge density of = -2 nC/m. What is the electric field (magnitude and direction) at

a) [5] r = 0.5 cm?

In a static situation all charge resides on surface of a conductor so for a Gaussian cylinder of radius 0.5 cm and length centred on the z axis

b) [5] r = 2.0 cm?

Consider a Gaussian cylinder of radius r = 2.0 cm and length centred on the z axis. Cylindrical symmetry dictates that points radially and its magnitude depends only on distance from the z axis. Hence,

c) [5] What is the surface charge density on the inner surface of the outer conductor?

Answer d) first; inside the outer cylinder (3 cm 4 cm) is zero, so the sum of the charge-per-unit-length of the inner conductor and the inner surface of the outer conductor must be zero. Hence,

d) [5] r = 3.5 cm?

In a static situation inside a conductor is identically zero.

e) [5] r = 5.5 cm?

Consider a Gaussian cylinder of radius r = 5.5 cm and length centred on the z axis. Cylindrical symmetry dictates that points radially and its magnitude depends only on distance from the z axis. Hence,

2) A 0.2 m long Nichrome wire of unknown resistance is connected between the terminals of a 3.00 V battery. Energy is dissipated in the wire at the rate of 0.54 W.

a) [7] What is the resistance of the Nichrome wire?

b) [6] The Nichrome wire is now disconnected from the battery and connected to a current source of 0.2 A. What is the power dissipation?

c) [7] What is the electric field within the wire in b)?

d) [5] Three identical 0.2 m long wires are twisted together to form a single 0.2 m long wire which is then connected to the 3.00 V battery from a). Calculate the power dissipation.

Resistance is proportional to 1/A, where A is the cross sectional area of the material carrying the current. Hence,

3) A parallel plate 10 nF capacitor is connected to a battery which keeps a potential difference of 30 V between the plates. If the distance, d, between the plates is doubled:

a) [6] What is the new capacitance?

b) [6] What is the change in potential energy stored in the capacitor?

c) [6] What is the change in the charge stored on the capacitor?

d) [7] If the battery is disconnected, how much work needs to be done to increase the distance between the plates from 2d to 3d? (Hint: it is the charge, Q, that is kept constant now.)

4) A circular (R = 8 cm) ring of charge Q = 10  C lies in the x-ŷ plane, centred on the origin. A point charge of -10  C lies on the z axis at z = -6 cm as shown.

a) [8] What is the electric potential at the origin due to the ring [V( ) = 0]?

where denotes the position of the charge dq. Hence,

b) [8] What is the total electric potential at the origin due to the ring and the point charge [V( ) = 0]?

Superposition allows one to add the potential due to the ring to the potential due to the point charge:

c) [9] Where on the z axis (in addition to z =  ) is the potential zero?

Distance from any point on the ring to a point z on the z axis is and the distance from the point charge to that point is . Hence, the potential at an arbitrary point z is given by:

Setting V to zero allows one to write: