In the problem 3.30 of Introduction to Electrodynamics by David J Griffiths, the task is to calculate the dipole moment for given charge distribution, which is a surface charge σ=kcosθ.
In the solution manual, the writer does this as follows:
Now, my question is why z has been used to calculate this dipole moment? According to previous discussion, I thought we should use r, but that is not the case I guess. Kindly help!
Dipole moment of sphere for given surface charge density
electrostatics
Best Answer
For a surface charge density $\sigma$, the dipole moment is:
$$ \mathbf{p} = \int \sigma \mathbf{r} \, \mathrm{d}a $$
This is a vector equation, which means that it is equivalent to a set of 3 equations involving the components of both sides:
$$ \begin{align} p_x &= \int \sigma x \, \mathrm{d}a \\ p_y &= \int \sigma y \, \mathrm{d}a \\ p_z &= \int \sigma z \, \mathrm{d}a \end{align} $$
The author of the solution has pointed out that $p_x$ and $p_y$ are zero and quickly moved on to calculating $p_z$.