Relation between electrostatic potential and field
Relation between Electrostatic Potential and Field
Electrostatics is a branch of physics that studies electric charges at rest. Two fundamental concepts in electrostatics are the electrostatic potential and the electrostatic field. Understanding the relationship between these two concepts is crucial for comprehending how electric charges interact in a static situation.
Electrostatic Field (E)
The electrostatic field (also known as the electric field) is a vector field that represents the force experienced by a positive test charge placed in the vicinity of other electric charges. It is defined as the force (F) per unit charge (q):
$$ \vec{E} = \frac{\vec{F}}{q} $$
The direction of the electric field at any point is the direction of the force that a positive test charge would experience if placed at that point.
Electrostatic Potential (V)
Electrostatic potential, on the other hand, is a scalar quantity that represents the potential energy per unit charge at a point in an electric field. It is the work done in bringing a unit positive charge from infinity to that point without acceleration:
$$ V = \frac{W}{q} $$
The potential is a measure of the potential energy that a charge would have due to its position in the electric field.
Relation between Electrostatic Potential and Field
The electrostatic field is related to the electrostatic potential by the negative gradient of the potential:
$$ \vec{E} = -\nabla V $$
This equation tells us that the electric field is the spatial rate of change of the electric potential. In other words, the electric field points in the direction of the greatest decrease of potential.
Important Points and Differences
| Electrostatic Field (E) | Electrostatic Potential (V) |
|---|---|
| Vector quantity | Scalar quantity |
| Has both magnitude and direction | Has only magnitude |
| Represented by field lines | Represented by equipotential surfaces |
| Measured in volts per meter (V/m) | Measured in volts (V) |
| Related to force experienced by a charge | Related to potential energy of a charge |
| Can do work on a charge | Work done is independent of the path taken |
Formulas
- Electric Field due to a point charge: $$ \vec{E} = \frac{kQ}{r^2} \hat{r} $$
- Electric Potential due to a point charge: $$ V = \frac{kQ}{r} $$
- Work done by the electric field: $$ W = -q \Delta V $$
- Relationship between E and V in one dimension: $$ E = -\frac{dV}{dx} $$
Examples
Example 1: Electric Field from Potential Gradient
Consider a region where the electric potential V is given by ( V(x) = -5x^2 ) volts, where x is in meters. The electric field at a point x can be found by taking the negative gradient of the potential:
$$ E(x) = -\frac{dV}{dx} = -\frac{d(-5x^2)}{dx} = 10x \text{ V/m} $$
Example 2: Potential Difference and Work Done
A charge of 2 C is moved from a point A, where the potential is 10 V, to a point B, where the potential is 5 V. The work done by the electric field is:
$$ W = -q \Delta V = -2 \times (5 - 10) = 10 \text{ J} $$
The positive work indicates that the electric field is doing work on the charge to move it from a higher potential to a lower potential.
Understanding the relationship between electrostatic potential and field is essential for solving problems in electrostatics and for grasping the fundamental principles of electric forces and energy.