Calculating Total Dipole Moment of Spherical Sheet

[hansbahia]

Homework Statement

A spherical sheet of radius a has surface-charge density which depends on the polar angle θ according to the formula

σ(θ)=σ0+σ1cos(θ)

Find the total dipole moment

Homework Equations

pz=∫(dQ)z
z=rcosθ

The Attempt at a Solution

I calculated the 1st dipole moment
pz=∫∫d∅dθσ0a^3sinθcos^2θ
pz=4/3pi*a^3σ0

I'm not understanding the question, "total" dipole moment?

 

[gabbagabbahey]

I'm not understanding the question, "total" dipole moment?

When you don't understand a term, the most logical thing to do is to look it up (first try looking in your textbook & notes, and if it isn't there, then try Google, and finally just ask your instructor for clarification).

Total dipole moment just means the electric dipole moment of the entire distribution (the full vector form, not just the z-component that you get from the 2 relevant equations you posted).

 

[hansbahia]

Thanks, I figured out. I had to integrate from 0 to pi

 

[gabbagabbahey]

Thanks, I figured out. I had to integrate from 0 to pi

But do you see why the x- and y-components of the dipole moment are zero here? Getting the right answer is not the same as solving the problem.

 

[paranormal]

The total dipole moment of a system is the vector sum of all individual dipole moments within that system. In this case, the spherical sheet can be considered as a collection of infinitesimal dipoles, each having a dipole moment of pz = σ(a)cosθ dA, where dA is the differential area element. The total dipole moment can then be calculated by integrating over the entire surface of the sheet:

P = ∫∫ pz dA = ∫∫ σ(a)cosθ dA

Using the given surface charge density, we can rewrite this as:

P = ∫∫ (σ0 + σ1cosθ)cosθ dA

Now, we need to express dA in terms of θ and integrate over the appropriate limits. Since the sheet is spherical, we can use the spherical coordinate system with limits of θ from 0 to π and φ from 0 to 2π. The differential area element in this coordinate system is given by dA = a^2sinθ dθdφ. Substituting this into our equation, we get:

P = ∫0^2π ∫0^π (σ0 + σ1cosθ)cosθ a^2sinθ dθdφ

Simplifying and solving the inner integral first, we get:

P = a^2∫0^2π σ0 sinθ dφ + a^2σ1 ∫0^2π cosθ^2 sinθ dφ

Using the trigonometric identity cos^2θ = (1+cos2θ)/2 and solving the second integral, we get:

P = a^2∫0^2π σ0 sinθ dφ + a^2σ1/2 ∫0^2π (1+cos2θ)sinθ dφ

Integrating and simplifying further, we get:

P = 4πa^2σ0 + 2πa^2σ1

Therefore, the total dipole moment of the spherical sheet is given by:

P = 4πa^2(σ0 + 1/2σ1)

This can also be written in terms of the total surface charge Q, which is given by Q = 4πa^2σ0, as:

P = Q + 1/2Qσ1

In summary, the total