Question

What is the relationship between electric field and electric potential due to point charge?

Answer 1

The relationship between electric field (E) and electric potential (V) due to a point charge is fundamental in electrostatics.

1. **Electric Field (E)**: The electric field created by a point charge \( Q \) at a distance \( r \) from the charge is given by the formula:
   \[
   E = \frac{k |Q|}{r^2}
   \]
   where \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)).

2. **Electric Potential (V)**: The electric potential due to a point charge at a distance \( r \) is given by:
   \[
   V = \frac{k Q}{r}
   \]

### Relationship

The electric field is the negative gradient of the electric potential:
\[
E = -\frac{dV}{dr}
\]

This means that the electric field points in the direction of decreasing potential. Specifically, for a positive charge, the potential decreases as you move away from the charge, leading to a positive electric field directed away from the charge. For a negative charge, the potential increases as you move away from the charge, resulting in an electric field directed toward the charge.

### Summary

- The electric field \( E \) is related to the change in electric potential \( V \) with respect to distance \( r \).
- The field indicates the direction and magnitude of the force experienced by a positive test charge, while the potential represents the work done per unit charge in bringing a charge from infinity to a point in space.

Answer 2

The electric field and electric potential due to a point charge are closely related concepts in electrostatics.

1. **Electric Field (E)**: This is a vector quantity that represents the force per unit charge experienced by a test charge placed in the field. For a point charge \( q \), the electric field \( \vec{E} \) at a distance \( r \) from the charge is given by:
   \[
   \vec{E} = \frac{k q}{r^2} \hat{r}
   \]
   where \( k \) is Coulomb's constant, and \( \hat{r} \) is a unit vector pointing radially away from the charge if \( q \) is positive, or towards the charge if \( q \) is negative.

2. **Electric Potential (V)**: This is a scalar quantity that represents the work done per unit charge in bringing a test charge from infinity to a point in the field. For a point charge \( q \), the electric potential \( V \) at a distance \( r \) from the charge is given by:
   \[
   V = \frac{k q}{r}
   \]

**Relationship**: The electric field is related to the electric potential by the gradient. Specifically, the electric field is the negative gradient of the electric potential:
   \[
   \vec{E} = -\nabla V
   \]
For a point charge, in spherical coordinates, this relationship simplifies to:
   \[
   E = -\frac{dV}{dr}
   \]
Substituting \( V = \frac{k q}{r} \), we get:
   \[
   E = -\frac{d}{dr} \left( \frac{k q}{r} \right) = \frac{k q}{r^2}
   \]

Thus, the electric field is the spatial rate of change of the electric potential.