Define electric potential due to a point charge and arrive at the expression for the electric potential at a point due to a point charge in its vicinity.

Electric potential, often denoted as \( V \), is a measure of the potential energy per unit charge at a specific point in an electric field created by a point charge. It indicates how much work would be needed to move a unit positive charge from a reference point (usually at infinity) to the point in question without any acceleration.

Understanding Electric Potential

The electric potential due to a point charge can be derived from the concept of electric field and the work done in moving a charge within that field. A point charge \( Q \) generates an electric field around it, and the potential at a distance \( r \) from the charge can be calculated.

Deriving the Expression

To derive the expression for electric potential \( V \) at a distance \( r \) from a point charge \( Q \), we start with the formula for electric field \( E \) due to a point charge:

• The electric field \( E \) at a distance \( r \) from the charge is given by:
• E = \frac{kQ}{r^2}, where \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)).

Next, we consider the work done \( W \) in moving a unit positive charge from infinity to the point at distance \( r \). The work done is related to the electric potential:

• W = -\int_{\infty}^{r} E \, dr
• Substituting the expression for \( E \):
• W = -\int_{\infty}^{r} \frac{kQ}{r^2} \, dr

Evaluating this integral gives:

• W = -\left[-\frac{kQ}{r}\right]_{\infty}^{r} = \frac{kQ}{r}

Since electric potential \( V \) is defined as the work done per unit charge, we have:

Final Expression for Electric Potential

Thus, the electric potential \( V \) at a distance \( r \) from a point charge \( Q \) is expressed as:

V = \frac{kQ}{r}

This formula indicates that the electric potential decreases with increasing distance from the charge and is directly proportional to the magnitude of the charge.