Relation Between Electric Field and Electric Potential
Understanding the relationship between electric field (E) and electric potential (V) is fundamental in electrostatics. They are intrinsically linked, describing the same electrical phenomenon from different perspectives. The electric field represents the force per unit charge, while electric potential represents the potential energy per unit charge.
The Gradient of Potential
The electric field can be derived from the electric potential. Specifically, the electric field is the negative gradient of the electric potential. In simpler terms, the electric field points in the direction of the steepest decrease in electric potential.
Electric field is the negative spatial rate of change of electric potential.
The electric field (E) is related to the electric potential (V) by the equation E = -∇V. This means the electric field is the negative of the gradient of the potential. The gradient is a vector that points in the direction of the greatest rate of increase of a scalar function, and its magnitude is that rate of increase.
Mathematically, in Cartesian coordinates, the gradient of a scalar function V(x, y, z) is given by ∇V = (∂V/∂x)i + (∂V/∂y)j + (∂V/∂z)k. Therefore, the electric field is E = -[(∂V/∂x)i + (∂V/∂y)j + (∂V/∂z)k]. This implies that the component of the electric field in any direction is equal to the negative of the rate of change of potential in that direction. For a one-dimensional case, if the potential only varies along the x-axis, then E_x = -dV/dx.
Potential Difference and Work Done
The work done by the electric field in moving a unit positive charge from point A to point B is equal to the negative of the change in electric potential (V_B - V_A). Conversely, the work done by an external agent against the electric field to move a unit positive charge from A to B is equal to the change in electric potential (V_B - V_A).
Think of potential as 'electrical height'. The electric field is like gravity, always pulling downhill (towards lower potential).
Consider a positive point charge at the origin. Its electric potential is given by V = kq/r. As you move away from the charge (increase r), the potential decreases. The electric field, E = kq/r² in the radial direction, also points radially outward, which is the direction of decreasing potential. The relationship E = -dV/dr holds true here, as dV/dr = d(kq/r)/dr = -kq/r². Thus, E = -(-kq/r²) = kq/r².
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Equipotential Surfaces
An equipotential surface is a surface on which the electric potential is constant. No work is done by the electric field when a charge moves along an equipotential surface. A key property is that the electric field lines are always perpendicular to the equipotential surfaces. This is because if there were a component of E along the surface, work would be done, contradicting the definition of an equipotential surface.
The electric field lines are always perpendicular to the equipotential surfaces.
Summary of Key Relationships
| Concept | Description | Mathematical Relation |
|---|---|---|
| Electric Field (E) | Force per unit charge; points in direction of decreasing potential. | E = -∇V |
| Electric Potential (V) | Potential energy per unit charge; scalar quantity. | V = -∫ E ⋅ dl |
| Work Done (W) | Work done by electric field moving charge q from A to B. | W = q(V_A - V_B) |
| Equipotential Surface | Surface where V is constant; E is perpendicular to it. | ∇V = 0 along the surface |
Learning Resources
Provides a clear explanation of the relationship between electric field and potential, including the gradient concept and work done.
A comprehensive overview of electric potential, its definition, units, and its relationship with the electric field.
Explains the connection between E and V using analogies and focusing on the concept of potential energy and work.
A video tutorial that visually explains how electric fields and potentials are related, including examples.
Details the mathematical derivation of the electric field from the electric potential, focusing on the gradient.
A lecture note that clearly illustrates equipotential lines and their relationship with electric field lines.
A blog post tailored for competitive exams like JEE, summarizing key concepts of electric field and potential.
A conceptual video explaining the gradient of a scalar field, which is crucial for understanding E = -∇V.
Explores the concepts of work, energy, and potential difference in the context of electric fields.
Covers the fundamental concepts of electric field and potential, including their interrelation and properties.