# Why Use z Instead of r for Dipole Moment Calculation?

• Idoubt
In summary, the problem is about finding the dipole moment of a spherical shell with charge distribution σ = kcosθ. The integral for dipole moment is defined as P= ∫r σ dζ, where r is the position of charge with respect to the origin. The solution manual uses Rcosθ instead of r, taking into consideration that the charge distribution has symmetry with respect to the x and y axes. This simplifies the integration process. However, it is important to note that the unit vector \hat{r} is a function of θ and \phi and should not be taken out of the integral. The integral can also be rewritten in Cartesian coordinates for easier integration.
Idoubt

## Homework Statement

I'm trying to do problem 3.28 in griffith's electrodynamics. The problem statement is, to find the dipole moment of a spherical shell with charge distribution σ = kcosθ

The way I tried to do it was to use the definition of dipole moment, which griffith defines as

P= ∫r σ dζ

where r = position of charge w.r.t origin ( in this case R ) and dζ is volume element.

The above integral gives 0 ( unless i did something stupid)

I looked up the solution manual and the way it does it is to use Rcosθ ie z instead of r. Can some1 explain why this is?

You should consider r in the integral as a vector. and since the charge distribution has symmetry with respect to x and y axes, we only consider z component of r which is Rcosθ. You can check x and y and make sure that they are zero.

I see now. My problem was that even though I looked at it as a vector, I didn't realize that the unit vector $\hat{r}$ itself was a function of θ and $\phi$ and I took it out of the integral. When i rewrite it in cartesian co-ordinates and do the integral for each component ( cartesian unit vectors are constant so I can take it out of the integral ) it comes out fine. But when I look at this, vector integration with spherical co-ordinates seems very complicated, is there an easier way than rewriting in cartesian co-ords and integrating?

i got confused with this problem too, thank you for taking it out.

## 1. What is the definition of dipole moment of a spherical shell?

The dipole moment of a spherical shell is a measure of the strength and direction of the electric dipole moment of the shell. It is defined as the product of the charge of the shell and the distance between the center of the shell and the center of the dipole.

## 2. How is the dipole moment of a spherical shell calculated?

The dipole moment of a spherical shell is calculated by multiplying the charge of the shell by the distance between the center of the shell and the center of the dipole. This can be represented by the equation: p = Q * d, where p is the dipole moment, Q is the charge of the shell, and d is the distance between the centers.

## 3. What is the direction of the dipole moment of a spherical shell?

The direction of the dipole moment of a spherical shell is determined by the orientation of the shell's charge. If the charge is evenly distributed, the dipole moment will be zero. If the charge is concentrated on one side, the dipole moment will point in the direction of the charge.

## 4. How does the dipole moment of a spherical shell affect electric fields?

The dipole moment of a spherical shell creates an electric field that is directed away from the positive charge and towards the negative charge. This electric field can interact with other charged particles and affect their movement and behavior.

## 5. Can the dipole moment of a spherical shell change?

Yes, the dipole moment of a spherical shell can change if the charge or the distance between the centers of the shell and the dipole changes. This change can also affect the strength and direction of the electric field created by the dipole moment.

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