In the problem discussed in the text, find the values of `E` and `V` at `p` dua to the remaining mass.

Explanation

To find the gravitational field \( E \) and gravitational potential \( V \) at point \( P \) due to the remaining mass, we will follow these steps:

### Step 1: Determine the mass of the cavity
The mass of the cavity can be calculated using the formula for the volume of a sphere and the density \( \rho \):
\[ \text{Mass of cavity} = \rho \times \text{Volume of cavity} = \rho \times \left( \frac{4}{3} \pi \left( \frac{R}{4} \right)^3 \right) \]
Calculating this gives:
\[ \text{Mass of cavity} = \rho \times \frac{4}{3} \pi \times \frac{R^3}{64} = \frac{\rho \pi R^3}{48} \]

### Step 2: Determine the mass of the solid sphere without the cavity
The mass of the solid sphere can be calculated similarly:
\[ \text{Mass of solid sphere} = \rho \times \left( \frac{4}{3} \pi R^3 \right) = \frac{4}{3} \pi R^3 \rho \]
Let’s denote this mass as \( M \).

### Step 3: Calculate the gravitational field \( E \)
The total gravitational field \( E \) at point \( P \) is the sum of the gravitational fields due to the solid sphere and the cavity:
\[ E = E_{\text{solid}} - E_{\text{cavity}} \]
Where:
- \( E_{\text{solid}} = -\frac{GM}{(3R)^2} \hat{i} \)
- \( E_{\text{cavity}} = -\frac{G \left( \frac{\rho \pi R^3}{48} \right)}{\left( \frac{9R}{4} \right)^2} \hat{i} \)

Calculating these values:
\[ E_{\text{solid}} = -\frac{G \left( \frac{4}{3} \pi R^3 \rho \right)}{9R^2} \hat{i} = -\frac{4G \pi R \rho}{27} \hat{i} \]
\[ E_{\text{cavity}} = -\frac{G \left( \frac{\rho \pi R^3}{48} \right)}{\left( \frac{9R}{4} \right)^2} = -\frac{G \left( \frac{\rho \pi R^3}{48} \right)}{\frac{81R^2}{16}} = -\frac{16G \rho \pi R}{3888} \]

### Step 4: Combine the gravitational fields
Now, substituting these values into the equation for \( E \):
\[ E = -\frac{4G \pi R \rho}{27} + \frac{16G \rho \pi R}{3888} \]

### Step 5: Calculate the gravitational potential \( V \)
The gravitational potential \( V \) at point \( P \) is given by:
\[ V = V_{\text{solid}} - V_{\text{cavity}} \]
Where:
- \( V_{\text{solid}} = -\frac{GM}{3R} \)
- \( V_{\text{cavity}} = -\frac{G \left( \frac{\rho \pi R^3}{48} \right)}{\frac{9R}{4}} \)

Calculating these values:
\[ V_{\text{solid}} = -\frac{G \left( \frac{4}{3} \pi R^3 \rho \right)}{3R} = -\frac{4G \pi R^2 \rho}{9} \]
\[ V_{\text{cavity}} = -\frac{G \left( \frac{\rho \pi R^3}{48} \right)}{\frac{9R}{4}} = -\frac{4G \rho \pi R^2}{432} \]

### Step 6: Combine the gravitational potentials
Now, substituting these values into the equation for \( V \):
\[ V = -\frac{4G \pi R^2 \rho}{9} + \frac{4G \rho \pi R^2}{432} \]

### Final Values
After performing the calculations, we find:
- Gravitational field \( E \) at point \( P \)
- Gravitational potential \( V \) at point \( P \)