THE UNIVERSITY OF BRITISH COLUMBIA

*Physics 401 *
Assignment #
**1:**

** REVIEW **

Wed. 04 Jan. 2006 - finish by Wed. 11 Jan.

This first assignment is just review, to make sure you haven't
forgotten (or can quickly recall) what you learned in PHYS 301/354
(or earlier) about the E&M covered in the first 7 chapters
of our textbook: David Griffiths, Wed. 04 Jan. 2006 - finish by Wed. 11 Jan.

**MAXWELL'S EQUATIONS:**- Starting with Maxwell's equations in differential form, derive Maxwell's equations in integral form.
- Starting with Maxwell's generalization of Ampère's Law, , derive the continuity equation, , which is the mathematical expression of charge conservation.
- Starting with Maxwell's equations in free space ( ), show that and each satisfy a wave equation. What is the speed of propagation of the resulting wave in each case?

**CHARGED CONDUCTORS:**Two spherical cavities, of radii*a*and*b*, are hollowed out from the interior of a solid neutral conducting sphere of radius*R*, as shown in the figure. There are charges*q*_{a}and*q*_{b}at the centres of the respective cavities.

**6.**If a third charge*q*_{c}were brought near the conductor, which (if any) would change:(

*i*) ?(

*ii*) ?(

*ii*) ?(

*iv*) The electric fields inside cavities*a*and*b*?(

*v*) The electric field outside the conductor?

**COAXIAL CAPACITOR:**A capacitor is constructed of two very long concentric cylindrical conductors with their common axis horizontal, as shown in the diagram. The space between them is exactly half filled with a linear dielectric liquid with dielectric constant .- Show that the electric field is radial and is the same in the dielectric half as in the vacuum half of the capacitor.
- Deduce the capacitance per unit length of this coaxial capacitor.
- If the conductors carry free charges per unit length ,
find the polarization in the dielectric
at any point a distance
*r*from the central axis, in terms of and*r*.

**LINEAR CURRENTS:**

**LAPLACE'S EQUATION:**Consider an infinitely long metal pipe, of radius*R*, which is placed at right angles to an otherwise uniform electric field .

Hint: Note that this situation has cylindrical symmetry (not spherical!), with no*z*dependence, and hence simplifies to a 2-D plane polar problem.

2005-12-24