In Ex. 3.9, we obtained the potential of a spherical shell with surface

chargeσ(θ)=kcosθ. In Prob. 3.30, you found that the field is pure dipole outside; it's uniforminside (Eq. 3.86). Show that the limit R0reproduces the deltafunction term in Eq. 3.106.

Short Answer

Expert verified

The volume integral of electric field due to a surface charge density σθ=kcosθover a sphere of radius R is given by -p3ε0where pis the dipole moment of the charge density.

Step by step solution

01

Step 1: Given data

The provided surface charge density isσθ=kcosθ

The radius of the sphere is R.

02

Potential of surface charge density and dipole moment of spherical shell

The potential of the provided surface charge density is given by

v=k3ε0rcosθ ....(1)

Here, ε0is the permittivity of free space, r and θare spherical polar coordinates.

Since rcosθ=z,the potential of the spherical shell can be written as

v=kz3ε0

Here, z is the Cartesian co-ordinate.

The dipole moment of the spherical shell is

p=4πR3k3z^....(2)

03

Electric field of spherical shell

The expression for the electric field is given as

E=-V

Substitute the value of the potential from equation (1) in the above equation.

localid="1657538019274" E=-kz3ε0=-k3ε0z^

From the expression of the dipole moment in equation (2)

p=4πR3k3z^kz^=3p^4πR3

Substitute the expression in the electric field equation and get the following

E=-13ε03p4πR3=-p4πε0R3

The electric field derived above blows up for R0. But the volume integral of the electric field gives

Edτ=E×43πR3

Here, dτis the infinitesimal volume element.

Substitute the expression for the electric field

localid="1657537878656" Edτ=-p4πε0R3×43πR3=-p3ε0

Thus, the delta function term from Eq. 3.106 is reproduced.

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Most popular questions from this chapter

In Ex. 3.9, we derived the exact potential for a spherical shell of radius R , which carries a surface charge σ=kcosθ.

(a) Calculate the dipole moment of this charge distribution.

(b) Find the approximate potential, at points far from the sphere, and compare the exact answer (Eq. 3.87). What can you conclude about the higher multipoles?

For the infinite slot (Ex. 3.3), determine the charge density σ(y)on

the strip at x=0, assuming it is a conductor at constant potential V0.

Find the charge density σ(θ) on the surface of a sphere (radius R ) that

produces the same electric field, for points exterior to the sphere, as a charge qat the point a<R onthe zaxis.

(a) Show that the quadrupole term in the multipole expansion can be written as

Vquad(r)=14πε01r3i,j-13ri^rj^Qij ............(1)

(in the notation of Eq. 1.31) where

Qij=12[3r'jr'j-(r')2δij]ρ(r')dτ' ..........(2)

Here

δij={10ifi=jifij ..........(3)

is the Kronecker Delta and Qijis the quadrupole moment of the charge distribution. Notice the hierarchy

Vmon=14πε0Qr;Vdip=14πε0rjpj^r2;Vquad(r^)=14πε01r3ij-13rirj^^Qij;......

The monopole moment (Q) is a scalar, the dipole moment p is a vector, the quadrupole moment Qij is a second rank tensor, and so on.

(b) Find all nine componentsQij of for the configuration given in Fig. 3.30 (assume the square has side and lies in the x-y plane, centered at the origin).

(c) Show that the quadrupole moment is independent of origin if the monopole and

dipole moments both vanish. (This works all the way up the hierarchy-the

lowest nonzero multipole moment is always independent of origin.)

(d) How would you define the octopole moment? Express the octopole term in the multipole expansion in terms of the octopole moment.

A charge is distributed uniformly along the z axis from z=-atoz=+a. Show that the electric potential at a point r is given by

Vr,θ=Q4πε01r1+13ar2P2cosθ+15ar4P4cosθ+...

for r>a.

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