If potential at the surface of a planet is taken as zero, the potential at infinty will be (`M` and `R` are mass of radius of the planet)

Options

A. zero
B. `oo`
C. `(GM)/(R )`
D. `-(GM)/(R )`

Answer

To find the potential at infinity when the potential at the surface of a planet is taken as zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Gravitational Potential**:
Gravitational potential \( V \) at a distance \( r \) from a mass \( M \) is given by the formula:
\[ V = -\frac{GM}{r} \]
where \( G \) is the gravitational constant.

2. **Defining the Problem**:
We are given that the potential at the surface of the planet (at radius \( R \)) is taken as zero. Therefore, we can express this as:
\[ V(R) = 0 \]

3. **Calculating Potential at the Surface**:
Using the formula for gravitational potential at the surface of the planet, we have:
\[ V(R) = -\frac{GM}{R} \]
Setting this equal to zero (as per the problem statement):
\[ -\frac{GM}{R} = 0 \]
This implies that the reference point for potential has changed.

4. **Finding Potential at Infinity**:
Since we have set the potential at the surface to zero, we can find the potential at infinity. The potential at infinity is traditionally considered to be zero when the surface potential is not adjusted. However, since we have set the surface potential to zero, we need to find the potential at infinity:
\[ V(\infty) = -\frac{GM}{\infty} = 0 \]
But since we have taken the potential at the surface as zero, we need to adjust for that. The work done to bring a unit mass from infinity to the surface must equal the change in potential energy:
\[ V(\infty) = V(R) + \frac{GM}{R} \]
Since \( V(R) = 0 \):
\[ V(\infty) = 0 + \frac{GM}{R} = \frac{GM}{R} \]

5. **Conclusion**:
Therefore, the potential at infinity when the potential at the surface of the planet is taken as zero is:
\[ V(\infty) = \frac{GM}{R} \]

### Final Answer:
The potential at infinity is \( \frac{GM}{R} \).

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