Relation between potential and intensity

Relation Between Electric Potential and Electric Field Intensity

The electric potential and electric field intensity are two core concepts in electrostatics. While electric potential (\( V \)) is a scalar quantity that represents potential energy per unit charge, electric field intensity (\( \vec{E} \)) is a vector quantity that describes the force experienced per unit charge. They are closely related through a spatial rate of change.

Mathematical Relation Between \( V \) and \( \vec{E} \)

Electric field intensity is defined as the negative gradient of electric potential. That is:

\( \vec{E} = -\nabla V \)

This means the electric field is oriented in the direction where the electric potential decreases most rapidly.

Step-by-Step Derivation

Consider a small test charge \( q \) moving through a displacement \( d\vec{x} \) in an electric field \( \vec{E} \). The infinitesimal work done by the electric field on the charge is:

\( dW = \vec{F} \cdot d\vec{x} = q \vec{E} \cdot d\vec{x} \)

From the definition of electric potential difference:

\( dV = \frac{dW}{q} = \vec{E} \cdot d\vec{x} \)

According to convention, the potential difference is taken as the negative of the work done by the electric field per unit charge.

\( dV = -\vec{E} \cdot d\vec{x} \)

For One-Dimensional Case

If the field varies only along the x-direction:

\( E_x = -\frac{dV}{dx} \)

In Vector Form (Three Dimensions)

\[ \vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \]

This is compactly written as: \( \vec{E} = -\nabla V \)

Physical Meaning

• The electric field is directed from regions of higher potential to lower potential.
• The magnitude of the electric field is proportional to how quickly the potential changes with position.
• If potential is constant (i.e., \( \nabla V = 0 \)), then \( \vec{E} = 0 \); this corresponds to an equipotential region.

Graphical Interpretation

In a graph of potential vs. position, the slope of the curve at any point gives the magnitude of the electric field at that point (with a negative sign). Steeper slopes indicate stronger fields.

Key Takeaways

• \( \vec{E} = -\nabla V \) is the fundamental relation connecting electric field and potential.
• Electric field is the spatial rate of change of potential with a negative sign.
• This relation helps in calculating one quantity from the other in problems involving electrostatics.