How is an electric field related with electrostatic potential?
2 Answers
The relationship between electric field (\( \mathbf{E} \)) and electrostatic potential (\( V \)) is fundamental in electrostatics. Here's how they are connected:
1. **Definition of Electric Field**: The electric field at a point in space is defined as the force per unit charge experienced by a small positive test charge placed at that point. Mathematically, it can be expressed as:
\[
\mathbf{E} = -\nabla V
\]
This equation states that the electric field is the negative gradient of the electrostatic potential. The negative sign indicates that the electric field points in the direction of decreasing potential.
2. **Relation to Potential Difference**: The electric field can also be expressed in terms of the potential difference (\( \Delta V \)) between two points (\( A \) and \( B \)):
\[
\mathbf{E} = -\frac{\Delta V}{d}
\]
where \( d \) is the distance between the two points. This means that the electric field is related to how quickly the potential changes with distance.
3. **Work Done**: The change in electrostatic potential energy when moving a charge in an electric field is given by:
\[
W = q \Delta V
\]
where \( W \) is the work done on a charge \( q \). This reinforces the idea that the electric field does work on charges as they move through different potentials.
In summary, the electric field is a vector quantity that represents the force on a charge, while the electrostatic potential is a scalar quantity that represents the potential energy per unit charge. The electric field can be derived from the potential by taking the gradient, indicating how potential changes in space.
1. **Definition of Electric Field**: The electric field at a point in space is defined as the force per unit charge experienced by a small positive test charge placed at that point. Mathematically, it can be expressed as:
\[
\mathbf{E} = -\nabla V
\]
This equation states that the electric field is the negative gradient of the electrostatic potential. The negative sign indicates that the electric field points in the direction of decreasing potential.
2. **Relation to Potential Difference**: The electric field can also be expressed in terms of the potential difference (\( \Delta V \)) between two points (\( A \) and \( B \)):
\[
\mathbf{E} = -\frac{\Delta V}{d}
\]
where \( d \) is the distance between the two points. This means that the electric field is related to how quickly the potential changes with distance.
3. **Work Done**: The change in electrostatic potential energy when moving a charge in an electric field is given by:
\[
W = q \Delta V
\]
where \( W \) is the work done on a charge \( q \). This reinforces the idea that the electric field does work on charges as they move through different potentials.
In summary, the electric field is a vector quantity that represents the force on a charge, while the electrostatic potential is a scalar quantity that represents the potential energy per unit charge. The electric field can be derived from the potential by taking the gradient, indicating how potential changes in space.