Two heavy spheres of mass m are kept separated by a distance 2r. The gravitational field and potential at the midpoint of the line joining the centres of the spheres are

Options

A. `(Gm^2)/(r^2)" and "0`
B. 0 and `-(2Gm)/r`
C. 0 and `(2Gm)/r`
D. none of these

Answer

To solve the problem of finding the gravitational field and potential at the midpoint of the line joining the centers of two heavy spheres of mass \( m \) separated by a distance \( 2r \), we can follow these steps:

### Step 1: Understanding the Setup
We have two spheres, both with mass \( m \), separated by a distance of \( 2r \). The midpoint between the two spheres is at a distance of \( r \) from each sphere.

### Step 2: Gravitational Field Calculation
The gravitational field \( g \) due to a mass \( m \) at a distance \( r \) is given by the formula:
\[ g = \frac{Gm}{r^2} \]
where \( G \) is the gravitational constant.

**For Sphere A (left sphere):**
- The distance from the midpoint to Sphere A is \( r \).
- The gravitational field due to Sphere A at the midpoint is:
\[ g_A = \frac{Gm}{r^2} \]
- This field points towards Sphere A.

**For Sphere B (right sphere):**
- The distance from the midpoint to Sphere B is also \( r \).
- The gravitational field due to Sphere B at the midpoint is:
\[ g_B = \frac{Gm}{r^2} \]
- This field points towards Sphere B.

### Step 3: Direction of Gravitational Fields
- The gravitational field \( g_A \) points towards Sphere A (to the left).
- The gravitational field \( g_B \) points towards Sphere B (to the right).

### Step 4: Net Gravitational Field
Since \( g_A \) and \( g_B \) are equal in magnitude but opposite in direction, we can express the net gravitational field \( g_{net} \) at the midpoint as:
\[ g_{net} = g_A - g_B = \frac{Gm}{r^2} - \frac{Gm}{r^2} = 0 \]
Thus, the net gravitational field at the midpoint is zero.

### Step 5: Gravitational Potential Calculation
The gravitational potential \( V \) due to a mass \( m \) at a distance \( r \) is given by:
\[ V = -\frac{Gm}{r} \]

**For Sphere A:**
- The potential at the midpoint due to Sphere A is:
\[ V_A = -\frac{Gm}{r} \]

**For Sphere B:**
- The potential at the midpoint due to Sphere B is:
\[ V_B = -\frac{Gm}{r} \]

### Step 6: Total Gravitational Potential
Since gravitational potential is a scalar quantity, we can simply add the potentials from both spheres:
\[ V_{total} = V_A + V_B = -\frac{Gm}{r} - \frac{Gm}{r} = -\frac{2Gm}{r} \]

### Final Results
- The gravitational field at the midpoint is:
\[ g_{net} = 0 \]
- The gravitational potential at the midpoint is:
\[ V_{total} = -\frac{2Gm}{r} \]