What is the rate of change of electric potential with respect to displacement called?
2 Answers
The rate of change of electric potential with respect to displacement is called the **electric field**.
Mathematically, if \( V \) represents the electric potential and \( \mathbf{E} \) represents the electric field, the relationship between the two is given by:
\[ \mathbf{E} = -\nabla V \]
where \( \nabla V \) is the gradient of the electric potential. In simpler terms, the electric field is the negative gradient of the electric potential, indicating how the potential changes with position in space.
Mathematically, if \( V \) represents the electric potential and \( \mathbf{E} \) represents the electric field, the relationship between the two is given by:
\[ \mathbf{E} = -\nabla V \]
where \( \nabla V \) is the gradient of the electric potential. In simpler terms, the electric field is the negative gradient of the electric potential, indicating how the potential changes with position in space.
The rate of change of electric potential with respect to displacement is called the **electric field**.
To understand this concept in detail:
1. **Electric Potential**: This is a measure of the potential energy per unit charge at a specific point in an electric field. It essentially describes how much potential energy a charged particle would have if placed at that point. The electric potential (V) at a point in space is defined by the formula:
\[
V = \frac{U}{q}
\]
where \( U \) is the electric potential energy and \( q \) is the charge.
2. **Electric Field**: The electric field (\( \mathbf{E} \)) is a vector quantity that represents the force experienced by a unit positive charge placed at a point in space. It is defined as the negative gradient of the electric potential. Mathematically, the electric field is given by:
\[
\mathbf{E} = -\nabla V
\]
where \( \nabla V \) represents the gradient of the electric potential \( V \). In simpler terms, the electric field is the rate of change of electric potential with respect to displacement in space, and it points in the direction where the potential decreases the fastest.
3. **Gradient of a Scalar Field**: The gradient (\( \nabla \)) of a scalar field like electric potential gives a vector field that points in the direction of the greatest rate of increase of that scalar field. In the case of electric potential, taking the negative gradient results in the electric field, which points in the direction of the greatest rate of decrease of the electric potential.
In summary, the electric field measures how the electric potential changes with position, and it is directly related to the force that a charge would experience in that field.
To understand this concept in detail:
1. **Electric Potential**: This is a measure of the potential energy per unit charge at a specific point in an electric field. It essentially describes how much potential energy a charged particle would have if placed at that point. The electric potential (V) at a point in space is defined by the formula:
\[
V = \frac{U}{q}
\]
where \( U \) is the electric potential energy and \( q \) is the charge.
2. **Electric Field**: The electric field (\( \mathbf{E} \)) is a vector quantity that represents the force experienced by a unit positive charge placed at a point in space. It is defined as the negative gradient of the electric potential. Mathematically, the electric field is given by:
\[
\mathbf{E} = -\nabla V
\]
where \( \nabla V \) represents the gradient of the electric potential \( V \). In simpler terms, the electric field is the rate of change of electric potential with respect to displacement in space, and it points in the direction where the potential decreases the fastest.
3. **Gradient of a Scalar Field**: The gradient (\( \nabla \)) of a scalar field like electric potential gives a vector field that points in the direction of the greatest rate of increase of that scalar field. In the case of electric potential, taking the negative gradient results in the electric field, which points in the direction of the greatest rate of decrease of the electric potential.
In summary, the electric field measures how the electric potential changes with position, and it is directly related to the force that a charge would experience in that field.