Two bodies of masses m and M are placed at distance d apart. The gravitational potential (V) at the position where the gravitational field due to them is zero V is

Options

A. `V=-(G)/(d)(m+M)`
B. `V=-(G)/(d)`
C. `V=-(GM)/(d)`
D. `V=-(G)/(d)sqrt(sqrt(m)+sqrt(M))^(2)`

Explanation

To solve the problem of finding the gravitational potential (V) at the point where the gravitational field due to two masses (m and M) is zero, we can follow these steps:

### Step 1: Understand the Setup
We have two masses, \( m \) and \( M \), placed at a distance \( d \) apart. We need to find a point between them where the gravitational field is zero.

### Step 2: Define the Position
Let’s denote the distance from mass \( m \) to the point where the gravitational field is zero as \( x \). Consequently, the distance from mass \( M \) to this point will be \( d - x \).

### Step 3: Write the Gravitational Fields
The gravitational field \( E \) due to mass \( m \) at the point is given by:
\[ E_m = -\frac{Gm}{x^2} \]
The gravitational field \( E \) due to mass \( M \) at the point is given by:
\[ E_M = -\frac{GM}{(d - x)^2} \]

### Step 4: Set the Gravitational Fields Equal
At the point where the gravitational field is zero, the magnitudes of the gravitational fields due to both masses must be equal:
\[ \frac{Gm}{x^2} = \frac{GM}{(d - x)^2} \]

### Step 5: Cross Multiply and Simplify
Cross-multiplying gives:
\[ Gm(d - x)^2 = GMx^2 \]
Cancelling \( G \) from both sides, we have:
\[ m(d - x)^2 = Mx^2 \]

### Step 6: Expand and Rearrange
Expanding the left side:
\[ m(d^2 - 2dx + x^2) = Mx^2 \]
Rearranging gives:
\[ md^2 - 2mdx + mx^2 - Mx^2 = 0 \]
This simplifies to:
\[ md^2 - 2mdx + (m - M)x^2 = 0 \]

### Step 7: Solve the Quadratic Equation
This is a quadratic equation in \( x \). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = m - M \), \( b = -2md \), and \( c = md^2 \).

### Step 8: Calculate the Roots
The roots are:
\[ x = \frac{2md \pm \sqrt{(-2md)^2 - 4(m - M)(md^2)}}{2(m - M)} \]
This simplifies to:
\[ x = \frac{2md \pm \sqrt{4m^2d^2 - 4m(m - M)d^2}}{2(m - M)} \]
\[ x = \frac{2md \pm 2d\sqrt{mM}}{2(m - M)} \]
\[ x = \frac{d(m + \sqrt{mM})}{m - M} \]

### Step 9: Find the Gravitational Potential
The gravitational potential \( V \) at the point where the field is zero is given by the sum of potentials due to both masses:
\[ V = -\frac{Gm}{x} - \frac{GM}{d - x} \]
Substituting \( x \) and \( d - x \) into this equation will yield the final expression for the gravitational potential at that point.

### Final Expression
After substituting and simplifying, we find:
\[ V = -\frac{G}{d} \left( \sqrt{m} + \sqrt{M} \right)^2 \]