Solution

The key to solving this problem is the **principle of superposition**. We can treat the cavity as a combination of a solid, uniformly charged plate (density $+\rho$) and a sphere of the same size with an opposite charge density ($-\rho$).

Field at Point A

In a solid infinite plate of thickness $h$, the electric field at the exact center (point $A$) is zero due to symmetry. Therefore, the field at $A$ is generated solely by the "negative" sphere.

Using Gauss’s Law for the sphere at its surface ($r = h/2$):
$$E_A \cdot 4\pi \left(\frac{h}{2}\right)^2 = \frac{Q_{enc}}{\varepsilon_0}$$

Given $V = \frac{4}{3}\pi\left(\frac{h}{2}\right)^3$ and $Q_{enc} = \rho \cdot V$:
$$E_A \cdot 4\pi \left(\frac{h}{2}\right)^2 = \frac{\rho \cdot \frac{4}{3}\pi\left(\frac{h}{2}\right)^3}{\varepsilon_0}$$

**Answer 1:**
$$E_A = \frac{\rho h}{6\varepsilon_0}$$

Field at Point B

Point $B$ is on the surface of the plate.

1. **Field from the solid plate ($E_p$):** By Gauss’s Law, the total field emerging from both sides of a plate is $\frac{\rho h}{\varepsilon_0}$. For just one side (at point $B$):
   $$E_{plate} = \frac{\rho h}{2\varepsilon_0}$$
2. **Field from the "negative" sphere ($E_s$):** Point $B$ is on the surface of the sphere ($r = h/2$). As calculated in Part 1, the field from the sphere at its surface is:
   $$E_{sphere} = \frac{\rho h}{6\varepsilon_0}$$

The vectors point in opposite directions at $B$, so we subtract them:
$$E_B = E_{plate} - E_{sphere} = \frac{\rho h}{2\varepsilon_0} - \frac{\rho h}{6\varepsilon_0}$$

**Answer 2:**
$$E_B = \frac{\rho h}{3\varepsilon_0}$$

Field along line OA

Since line $OA$ is at the center of the infinite plate, the plate's own contribution to the field is zero at every point along this line. Thus, the field depends only on the charged sphere.

For any point at a distance $r$ inside the cavity ($0 \leq r \leq h/2$), we apply Gauss's Law to the sphere:
$$E(r) \cdot 4\pi r^2 = \frac{\rho \cdot \frac{4}{3}\pi r^3}{\varepsilon_0}$$

**Answer 3:**
$$E(r) = \frac{\rho r}{3\varepsilon_0}$$