Two spheres one of mass `M` has radius `R`. Another sphere has mass `4M` and radius `2R`. The centre to centre distance between them is `12 R`. Find the distance from the centre of smaller sphere where
(a) net gravitational field is zero,
(b) net gravitational potential is half the potential on the surface of larger sphere.

Explanation

To solve the problem, we will break it down into two parts as stated in the question.

### Part (a): Finding the distance from the center of the smaller sphere where the net gravitational field is zero.

1. **Define the Variables**:
- Let the mass of the smaller sphere be \( M \) and its radius be \( R \).
- Let the mass of the larger sphere be \( 4M \) and its radius be \( 2R \).
- The distance between the centers of the two spheres is \( 12R \).
- Let \( x \) be the distance from the center of the smaller sphere to the point where the gravitational field is zero.

2. **Determine the Distance from the Larger Sphere**:
- The distance from the center of the larger sphere to the point \( P \) will be \( 12R - x \).

3. **Gravitational Fields**:
- The gravitational field \( E \) due to the smaller sphere at point \( P \) is given by:
\[ E_1 = -\frac{GM}{x^2} \]
(acting towards the smaller sphere).
- The gravitational field \( E \) due to the larger sphere at point \( P \) is given by:
\[ E_2 = -\frac{4GM}{(12R - x)^2} \]
(acting towards the larger sphere).

4. **Setting the Gravitational Fields Equal**:
- For the net gravitational field to be zero at point \( P \):
\[ \frac{GM}{x^2} = \frac{4GM}{(12R - x)^2} \]
- Cancel \( GM \) from both sides:
\[ \frac{1}{x^2} = \frac{4}{(12R - x)^2} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ (12R - x)^2 = 4x^2 \]

6. **Expanding and Rearranging**:
- Expanding the left side:
\[ 144R^2 - 24Rx + x^2 = 4x^2 \]
- Rearranging:
\[ 144R^2 - 24Rx - 3x^2 = 0 \]

7. **Using the Quadratic Formula**:
- This is a quadratic equation in the form \( ax^2 + bx + c = 0 \) where \( a = -3, b = -24R, c = 144R^2 \).
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{24R \pm \sqrt{(-24R)^2 - 4 \cdot (-3) \cdot 144R^2}}{2 \cdot (-3)} \]
- Simplifying:
\[ x = \frac{24R \pm \sqrt{576R^2 + 1728R^2}}{-6} \]
\[ x = \frac{24R \pm \sqrt{2304R^2}}{-6} \]
\[ x = \frac{24R \pm 48R}{-6} \]

8. **Calculating the Values**:
- This gives two solutions:
\[ x = \frac{72R}{-6} = -12R \quad \text{(not valid, as distance cannot be negative)} \]
\[ x = \frac{-24R}{-6} = 4R \]

9. **Conclusion for Part (a)**:
- The distance from the center of the smaller sphere where the net gravitational field is zero is \( x = 4R \).

### Part (b): Finding the distance from the center of the smaller sphere where the net gravitational potential is half the potential on the surface of the larger sphere.

1. **Gravitational Potential at Point P**:
- Let \( y \) be the distance from the center of the smaller sphere to point \( P \).
- The distance from the center of the larger sphere to point \( P \) is \( 12R - y \).

2. **Gravitational Potential Calculation**:
- The gravitational potential \( V \) due to the smaller sphere at point \( P \):
\[ V_1 = -\frac{GM}{y} \]
- The gravitational potential \( V \) due to the larger sphere at point \( P \):
\[ V_2 = -\frac{4GM}{12R - y} \]

3. **Total Gravitational Potential**:
- The total gravitational potential at point \( P \):
\[ V = V_1 + V_2 = -\frac{GM}{y} - \frac{4GM}{12R - y} \]

4. **Setting the Equation**:
- We need this potential to be half the potential on the surface of the larger sphere:
\[ V = -\frac{4GM}{2R} = -\frac{2GM}{R} \]
- Therefore, we set:
\[ -\frac{GM}{y} - \frac{4GM}{12R - y} = -\frac{2GM}{R} \]

5. **Simplifying the Equation**:
- Cancel \( GM \):
\[ -\frac{1}{y} - \frac{4}{12R - y} = -\frac{2}{R} \]
- Rearranging gives:
\[ \frac{1}{y} + \frac{4}{12R - y} = \frac{2}{R} \]

6. **Finding a Common Denominator**:
- Multiply through by \( y(12R - y)R \):
\[ (12R - y)R + 4y = 2y(12R - y) \]

7. **Expanding and Rearranging**:
- Expanding:
\[ 12R^2 - yR + 4y = 24Ry - 2y^2 \]
- Rearranging gives:
\[ 2y^2 - 9Ry + 12R^2 = 0 \]

8. **Using the Quadratic Formula**:
- Using \( a = 2, b = -9R, c = 12R^2 \):
\[ y = \frac{-(-9R) \pm \sqrt{(-9R)^2 - 4 \cdot 2 \cdot 12R^2}}{2 \cdot 2} \]
- Simplifying:
\[ y = \frac{9R \pm \sqrt{81R^2 - 96R^2}}{4} \]
\[ y = \frac{9R \pm \sqrt{-15R^2}}{4} \]
- This indicates that the solutions for \( y \) are complex, suggesting that the gravitational potential cannot reach half the value of the potential on the surface of the larger sphere at any point in the space between the two spheres.

### Conclusion for Part (b):
- The distances from the center of the smaller sphere where the net gravitational potential is half the potential on the surface of the larger sphere do not exist in the given configuration.