THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 2:
 
ELECTRIC  FIELDS  &  POTENTIALS
 
18 Jan. 1999 - finish by 25 Jan. 1999

1.

Lines of Charge: An infinite line of charge with linear charge density $\lambda = 5 \times 10^{-10}$ C/m lies along the x-axis (y=0 and z=0 from $x \to -\infty$ to $x \to +\infty$). A second infinite line of charge with exactly the opposite charge density lies along the z axis. What are the x, y and z components of the resultant electric field at the point (x,y,z) = (4,4,4) m?

2.

Triangle of Charges: Derive an expression for the work required to bring four charges together into an equilateral triangle of side a (as shown) with one charge at the centre of the triangle. (Initially the charges are all infinitely far apart.)

3.

Hole in a Plane of Charge: A large flat non-conducting surface carries a uniform charge density $\sigma = 4.0 \times 10^{-9}$ C/m². A small circular hole has been cut out of the middle of this sheet of charge, as shown on the diagram. Ignoring ``fringing'' of the field lines around all edges, calculate the electric field at point P a distance z = 1.0 m up from the centre of the hole along its axis. The radius of the hole is R = 0.4 m.

4.

Atoms as Spheres of Charge: In Rutherford's work on $\alpha$ particle scattering from atomic nuclei, he regarded the atom as having a pointlike positive charge of +Ze at its centre, surrounded by a spherical volume of radius R filled with a uniform charge density that makes up a total charge -Ze. In this simple model, show that the electric field strength E and the electric potential V at a distance r < R from the centre are given by

$$E(r<R) = \left( Ze \over 4\pi\varepsilon_0 \right) \left( {1 \over r^2} - {r \over R^3} \right) \quad \hbox{\rm and}$$

$$V(r<R) = \left( Ze \over 4\pi\varepsilon_0 \right) \left( {1 \over r} - {3 \over 2r} + {r^2 \over 2 R^3} \right) .$$

(We choose V to be zero at $r \to \infty$.)