Week 2

Basic Facts

The field E at a point r caused by a distributed charge density ρ(r) is given by

where r' points from the origin to the element of volume dV' and the integration is over all the space containing the charge.
If the charge is distributed over a plane then replace ρ(r')dV' by σ(r')da'. Similarly, for a line of charge replace ρ(r')dV' by λ(r')dV'.

Field lines are a common way to represent vector fields. They obey the rules
1) Lines are everywhere parallel to the field.
2) Lines start and end only on charges.
3) Field lines are more closely spaced in regions of high field and further apart in regions of low field.
4) Field lines are elastic (stretchy, but don't like it), flexible (bendy, but don't like it) and repulsive (they repel each other as if they were all positively charged).

The Electric Potential difference between two points r1 and r2 is defined to be

where the integral is along a line joining r1 to r2.
If the E field is caused only by static charges then the value of the integral does not depend on the path taken and the potential is unique. In that case we say that the field E is conservative.
In many situations we use an absolute electric potential that is defined by specifying a unique place at which the potential is set to zero. This is commonly the point at infinity.

The electric potential at point r arising from a set of charges Q_i at positions r_i is given by
V(r) = ∑_i        Q_i          
                4πε₀|r - r_i|
Note that if the charge distribution is continuous then the sum will become an integral over the volume containing the charge.

Useful Facts

The field from an infinite straight line of charge with density λ Coulombs/metre depends only on the perpendicular distance r from the line and is given by
E = __λ__ρ__
       2πε₀ |ρ|²