Week 3

Basic Facts

The electric potential around a point charge of magnitude Q depends only on the distance from the charge. Assuming that the potential at infinity is set to zero then the potential at a point r due to a charge Q at point r' is given by
V(r) =        Q       
            4πε₀|r - r'|

We typically represent the electric potential by drawing lines (in 2-D, surfaces in 3-D) along which the potential is a constant. These are called Equipotential lines (or surfaces). At every point the equipotentials are perpendicular to the lines of the electric field and the field points from the higher potential side to the lower. For example, here are the field and potential around an electric dipole.

In a region where the electric potential V(r) is known, we can calculate the electric field E(r) using the relation

where grad V(r) is given by

The work required to move a charge q from r₁ to r₂ = q(V(r₁) - V(r₁)).

If a particle of charge q moves freely from r₁ to r₂ then its K.E. changes by q(V(r₁) - V(r₂))

Two conductors with charges +Q and -Q form a Capacitor. There is a potential V between the conductors and the potential is proportional to the charge. We define the Capacitance by the equation
Q = CV so that C=Q/V.

When a capacitor is charged to a voltage V, charge Q = CV, it has stored energy
U = C V²/2 = Q²/(2C).

In general, a region of space containing an electric field E has an electrostatic energy density = ε₀E²/2.

The value of the capacitance depends only on the geometry of the conductors.
For a pair of parallel plates of area A with separation d,
C=ε₀A
       d
where d is very small compared to the size of the plates.

The field between the plates of such a parallel plate capacitor is uniform and has strength
E = σ
      ε₀