19. Remarks on Some Azimuthally Symmetric Problems
Jackson revisits our old friend, the split sphere with the top and bottom halves at respectively. Recall we found the potential on the axis as
where here we’ve replaced with Since we’re only in the region outside the sphere, we can rearrange to get a factor
Going off axis is now trivial: from the uniqueness theorem, we know the answer must be
(For this particular case of the hemispheres, only even powers of will enter, hence only odd powers of remember the charge distribution itself has odd parity.)
On the sphere itself, this polynomial expansion must equal the function equal to one on the top half hemisphere, minus one on the bottom half.
Therefore, the same series of coefficients must represent the potential inside the sphere, except that now is replaced by
General Axially Symmetric Charge Distribution on a Spherical Surface
Suppose we have a spherical surface with charge density axially symmetric. This will cause a discontinuity in the radial electric field at the surface, but not in the potential.
The coefficients must match to ensure the continuity of the potential, but the slope is discontinuous:
The orthogonality of the Legendre polynomials then gives
Ring of Charge
Jackson's example is a ring of charge along latitude total charge
The charge density function
Therefore, the potential from the ring is:
Note that as always the field with a certain angular symmetry is generated by a charge distribution having that symmetry.
Conical Hole or Sharp Point
This is discussed in Jackson: the problem is the solution of with on a conical surface. Again we have azimuthal symmetry, the axis of the cone is and Legendre's equation is still valid. However, we must now have for all points having the angle the angle of the cone.
Noticing from the graph below that the closest zero to gets closer and closer as increase, we can put it at if we choose a non-integral (in general) value of a solution of the form with the right boundary condition,
This generalized solution of Legendre's equation (and hence of ) is called a Legendre function of the first kind of order
From the graph, if degrees, so in the conical hole the field neat the origin will be approximately of order