A charge `Q` is placed at the origin. The electric potential due to this charge at a given point in space is `V`. The work done by an external force in bringing another charge q from infinity up to the point is

Options

A. `(V)/(q)`
B. `Vq`
C. `V+q`
D. `V`

Answer

To solve the problem, we need to determine the work done by an external force in bringing a charge \( q \) from infinity to a point where there is a charge \( Q \) placed at the origin. The electric potential \( V \) at that point due to charge \( Q \) is given.

### Step-by-Step Solution:

1. **Understanding Electric Potential**:
The electric potential \( V \) at a distance \( r \) from a point charge \( Q \) is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant. However, in this case, we are given that the potential at the point is \( V \).

2. **Work Done by External Force**:
The work done \( W \) by an external force in bringing a charge \( q \) from infinity (where the potential is considered zero) to a point where the potential is \( V \) can be calculated using the relationship:
\[ W = qV \]
This is because work done is equal to the charge multiplied by the potential difference (which is \( V - 0 = V \)).

3. **Substituting Values**:
Since we know \( V \) (the electric potential at the point due to charge \( Q \)) and the charge \( q \), we can substitute these values into the equation:
\[ W = qV \]

4. **Final Result**:
Therefore, the work done by the external force in bringing the charge \( q \) from infinity to the point is:
\[ W = qV \]

### Summary:
The work done by an external force in bringing a charge \( q \) from infinity to a point where the electric potential is \( V \) due to charge \( Q \) is given by:
\[ W = qV \]