Week 4 Lex 2: Conductors and Multipole Expansion.

Conductors: In this lecture we first examine the issue of charged and uncharged conductors in an electric field. A perfect ideal conductor is characterized by an unlimited supply of free electrons. When a conductor is placed in an external electric field E ₀ the electrons which are very loosely bound to the atom flow in the direction opposite to that of the external electric field. This causes separation of the positive and negative charges within the conductor and this sets up an induced Electric field E_ind inside the conductor in opposition to that of the external field. The charges flow till the induced field completely cancels the external applied field inside the conductor. Hence the field inside a conductor is ZERO and all charges appear on the surface of the conductor. ( inside the conductor the charge density is ZERO. This is because if any chrage density builds up inside the conductor, the resultant electric field with drive the charges till the density will is zero.)


Since the field is zero and the field is derivative of the potential E=-∇V so the conductor must be an equipotential surface with a constant potential that extends into the conductor. So on the surface of the conductor and inside the potential is constant. This also shows that the electric field has a discontinuity ( increases to a finite value from zero discontinuously) just away from the surface of the conductor. This always happens whenever the electric field encounters a surface charge density.


Now since dV=∇V.dl=0 on the surface as V is constant on the surface, this shows that the electric field just away from the surface of a conductor is normal to the surface ( -∇V is perpendicular to dl on the surface to make dV=0 on the surface which is an
equipotential).


So whenever a charge q is brought near a conductor the electric field draws the electrons to the side closer to the charge leaving the positive charges piled up on the side which is further away. Hence an induced charge distribution occurs on the surface of the conductor due to the presence of the charge q. As the conductor was originally neutral and since the field inside the conductor is zero, application of Gauss theorem shows that the induced surface charge is always equal to -q.


The fact that a conducting surface is an equipotential is used in a Capacitor which is an arrangement of two or more conductors at different potentials with charges +Q and -Q. The charge Q=CV where C is the capacitance of the arrangement. Note that the
capacitance is a geometrical quantity and depends on the shape size and arrangement of the conductors. The Capacitor when charged by a battery has electrons removed from the +ve plate to the -ve plate and this process continues till the electric field due to the plate charges exactly balances the electric field due to the battery. The work done to move the electrons is stored in the capacitor electrostatic (potential) energy. This energy will appear when the conductors are shorted through sparking. The energy expression may be calculated and W=1/2CV²


Multipole Expansion: In what we have seen over the last 3 weeks it seems that the fundamental problem of electrostatics which was to find the electric field due to a certain charge
distribution/configuration is far easier to obtain provided we know the potential due to this charge distribution at the field point since E=-∇ V and differentiation is easier than integration. However its not easy to find the potential due to a charge distribution because that also involves solving an integral although the integrand in this case is a scalar. Although possible in simple cases its not easy to do this integral for an arbitrary chrage distribution.


The multipole expansion is a way around this difficulty to obtain at least the approximate potential at a field point far way from the charge distribution. We will see that the approximate potential is a very good estimate and becomes better as more and more terms in the expansion ( series) are considered.


A single point charge is called a monopole and we know the potential due to this. A simple binomial expansion provides us with the approximate potential due to a charge pair called a physical dipole. This provides us with the idea of expanding the potential due an arbitrary localized charge distribution at a field point far away from the charge configuration. A binomial expansion an the approximation that the point at which the potential is required is far away provides us with a systematic expansion of the potential V(r) in terms of a series in powers of (1/r) and specific basic charge configurations like a monopole, dipole, quadrupole, octupole etc. The approximation becomes better and better as more terms in the series are included. Note that each successive term contributes less and less to the potential as powers of (1/r). Schematically the series may be written as
V(r)=1/4πε₀ [ K₀/r + K₁/r ² + K₂/r ³ +...........]
Where the terms denote the monopole, dipole and quadrupole contributions respectively. Note that the dipole term in the monopole expansion is different from the potential due to a physical dipole.
We will have more to say about this in the next lecture.