Animated Solution for Physics - Electrostatics: Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region

Creating a Uniform Electric Field

Imagine a fascinating scenario in electrostatics: two solid, non-conducting spheres, one packed uniformly with positive charge (density $+\rho$) and the other with negative charge (density $-\rho$). Now, push them together so they partially overlap. What does the electric field look like in that shared, overlapping space?

At first glance, it seems like a chaotic mess of vectors pointing in all directions. However, the reality is one of the most elegant results in physics.

To unravel this mystery, we must first recall a powerful consequence of Gauss's Law. For a uniformly charged solid sphere, the electric field at any point inside it is directly proportional to the distance from the center.

Mathematically, if a point has a position vector $\vec{r}$ relative to the center of the sphere, the electric field $\vec{E}$ is given by:

This simple, linear relationship is the key to unlocking our overlapping spheres problem.

Let's pick an arbitrary point $P$ inside the overlapping region. By the principle of superposition, the net electric field at $P$ is simply the vector sum of the electric fields produced by each sphere individually, as if the other didn't exist.

Let the center of the positive sphere be $C_1$ and the center of the negative sphere be $C_2$. The position vector of $P$ from $C_1$ is ${\vec{r}}_1$, and from $C_2$ is ${\vec{r}}_2$.

The electric field at $P$ due to the positive sphere is:

The electric field at $P$ due to the negative sphere (remembering its density is $-\rho$) is:

Adding these together gives us the net electric field:

Now, look closely at the geometry. We have a triangle formed by the points $C_1$, $C_2$, and $P$. Let's define a new vector, $\vec{d}$, which points from the center of the first sphere to the center of the second sphere ($\vec{d}=\overrightarrow{C_1{C}_2}$).

According to the triangle law of vector addition, if we travel from $C_1$ to $C_2$ (vector $\vec{d}$), and then from $C_2$ to $P$ (vector ${\vec{r}}_2$), it is equivalent to traveling directly from $C_1$ to $P$ (vector ${\vec{r}}_1$).

Rearranging this, we find a beautiful geometric truth:

Substitute this geometric identity back into our net electric field equation:

Look at this final expression! The position vectors ${\vec{r}}_1$ and ${\vec{r}}_2$ have completely vanished. The net electric field depends only on the charge density $\rho$ and the vector $\vec{d}$ connecting the two centers.

Since $\vec{d}$ is a constant vector (the centers of the spheres are fixed), the electric field is perfectly uniform in both magnitude and direction everywhere inside the overlapping region. This elegant principle is not just a textbook curiosity; it is the foundational model for understanding how dielectric materials polarize under external electric fields!