THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 122 Assignment # 4:
 
GAUSS' LAW
 
Fri. 25 Jan. 2002 - finish by Fri. 1 Feb.

1.

GAUSS' LAW FOR A SHEET OF CHARGE:  Imagine an infinite plane sheet of electric charge with $dQ/dA = \sigma$ units of charge per unit surface area. Using your own words and drawings,¹

(a)

Using only (i) the SUPERPOSITION PRINCIPLE for the total electric field due to an assembly of electric charges, plus (ii) simple SYMMETRY arguments, deduce the direction of the electric field due to the plane of charge.

(b)

Using the preceding result plus the general ideas of GAUSS' LAW and simple geometry, deduce the dependence of the magnitude E of the electric field upon x, the perpendicular distance away from the plane.

(c)

Show that your result agrees with the x-dependence of the electric field on axis due to a uniform disc of charge when $x \ll R$, the radius of the disc.

2.

FIELD WITHIN A UNIFORM CHARGE DISTRIBUTION: The textbook shows how to use GAUSS' LAW to derive the radial (r) dependence of the electric field E(r>R) outside charge distributions of spherical, cylindrical or planar symmetry, where R is the distance the charge distribution extends from the centre of symmetry - the radius of a charged sphere or cylinder, or half the thickness of an infinite slab of charge, respectively. Use similar arguments to show that, for each of these cases (a sphere, cylinder or a slab of uniform charge density), the electric field E(r<R) inside the charge distribution is given in terms of the field E(R) at the boundary of the charge distribution by

$$E(r<R) = \left( r \over R \right) E(R) .$$

3.

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, calculate the electric field strength E and the electric potential $\phi$ as functions of radius r and various constants. Plot your results for $0 < r \le 2R$. (Choose $\phi \goestoas{r \to \infty} 0$.)