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

2. **Potential at the Surface of the Planet**:
If we consider a planet with mass \( M \) and radius \( R \), the potential at the surface of the planet (at distance \( R \)) is:
\[ V_{\text{surface}} = -\frac{GM}{R} \]

3. **Setting the Reference Point**:
According to the problem, we are taking the potential at the surface of the planet as zero:
\[ V_{\text{surface}} = 0 \implies -\frac{GM}{R} = 0 \]
This means we are redefining our reference point for potential.

4. **Calculating Potential at Infinity**:
The potential at infinity is typically considered as \( V_{\infty} = 0 \). However, since we are redefining the potential at the surface to be zero, we need to find the potential at infinity relative to this new reference point.

5. **Using the Potential Difference**:
The potential difference between the surface of the planet and infinity can be expressed as:
\[ V_{\text{surface}} - V_{\infty} = -\frac{GM}{R} \]
Since \( V_{\text{surface}} = 0 \), we can substitute this into the equation:
\[ 0 - V_{\infty} = -\frac{GM}{R} \]
Rearranging gives us:
\[ V_{\infty} = \frac{GM}{R} \]

6. **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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