TOPIC 3 Gauss’s Law

Introduction From last lecture: Electric field strength is equal to the density of field lines Number of lines starting on a charge is proportional to the charge Gauss’s Law relates flux (= flow of field lines) through a closed surface to total charge enclosed Very useful for calculating field, especially for symmetric charge distributions.

Electric Flux Consider surface, area A, in uniform field E Flux  is field  projected area  = E A cos Can also consider this as area  normal component of field. Define vector A, magnitude proportional to area, direction of normal to the surface.  = E.A

For non-uniform field: Note: lines passing one way through surface give positive contribution to , other direction negative. Continuous lines  net flux through closed surface = 0

Consider spherical Gaussian surface around point charge: By Coulomb’s Law: So What about a different shape surface? Net flux through combined dotted and dashed surfaces = 0. Flux through any surface surrounding Q is Q/0.

Gauss’s Law The net flux through any closed surface around a total charge Qtot is just Qtot/0. Using Gauss’s Law to determine the field: Use the symmetry of the charges to determine the pattern of field lines. Choose a Gaussian surface which is locally either parallel to E or perpendicular to it. Where E is parallel to dA, E is locally constant.

Example 1 Show that when an isolated conductor carries a charge, it must reside only on the surface. Example 2 Show that the field about a charged conducting sphere is identical to that about the same charge concentrated at the centre of the sphere. Example 3 A charge Q is distributed uniformly throughout an insulating sphere of radius R. What is the electric field at points (a) outside the sphere (b) inside the sphere?

Example 4 An infinite sheet carries a surface charge density . What electric field does it produce? Example 5 An infinitely long wire carries a linear charge density . How does the electric field it produces vary with distance from the wire?