Electric field from the intersection of two spheres with different charges

Question

This question is repeated a lot, but I for two spheres with different charge densities, sphere one with radius a centered at the origin and charge density $\rho_1$, and sphere two with radius centered at a point d on the x axis a and charge density $\rho_2$, and their intersection is has a charge density $\rho_1+\rho_2$ i.e:

\begin{align} \rho(\vec{r}) &=\rho_1 \;\;\;\;\;\;\;\;\text{for}\; |\vec{r}| \leq a \; \text{and} \; |\vec{r}-d \hat{e_x}| > a \\ &=\rho_2 \;\;\;\;\;\;\;\;\text{for}\; |\vec{r}| > a \; \text{and} \; |\vec{r}-d \hat{e_x}| \leq a \\ &=\rho_1+\rho_2 \;\text{for}\; |\vec{r}| \leq a \; \text{and} \; |\vec{r}-d \hat{e_x}| \leq a \\ &=0 \;\;\; \text{else} \end{align}

My idea was to assume that both spheres have the charge $\rho_1+\rho_2$, calculate the electric field using Gauss Law for both sphere, and then use super position principle to find the field inside the intersection, anyhow I am not completely sure if I am allwoed to use this assumption that both spheres are charged with $\rho_1+\rho_2$, so can I make this assumption? or should I assume that each sphere is only charges with $\rho_1$ or $\rho_2$ respectively and then use the superposition principle?

Answer 1

Yes, you can assume that each sphere is only charged with $ρ_1$ and $ρ_2$ respectively and then use the superposition principle (justification below).

Superposition principle allows you to do a lot of interesting things with electric fields and potentials.

First, you can sum up the fields from all partitions of the initial charge and you will still get the total field.

Also, since the electric fields and potentials are linear with respect to charge, you can even split the charge on one partition to smaller amounts.

So, in this case, you can look at this problem as if you have 2 distinct charged spheres with $\rho_1$ and $\rho_2$. Now, the amazing thing is, even though these two have an intersection, thanks to superposition, we can split the current density in that region ($\rho_1 + \rho_2$) as one region with $\rho_1$ and one region with $\rho_2$. Then you can think the $\rho_1$ part along with the rest of $\rho_1$ as a full sphere, and the same for $\rho_2$.

The idea is to get rid of part of the charges and calculate fields, and so on until every charge is accounted for. The benefit of using sphere charges is that the electric fields of these can be very easily calculated using Gauss' law, so it's only convenient. Any other partitioning would be equally correct.