Derive the relation between electric field and electric potential due to a point charge.

The relationship between electric field (E) and electric potential (V) due to a point charge can be derived from fundamental principles of electrostatics.

Key Concepts

• Electric Field (E): A vector quantity that represents the force per unit charge experienced by a positive test charge placed in the field.
• Electric Potential (V): A scalar quantity that indicates the potential energy per unit charge at a point in an electric field.

Electric Field of a Point Charge

The electric field created by a point charge (Q) at a distance (r) from the charge is given by the formula:

E = k * |Q| / r²

Here, k is Coulomb's constant, approximately equal to 8.99 x 10⁹ N m²/C².

Electric Potential of a Point Charge

The electric potential at a distance (r) from a point charge (Q) is expressed as:

V = k * Q / r

Deriving the Relation

The electric field is related to the electric potential through the gradient of the potential. Mathematically, this is represented as:

E = -dV/dr

To find the relationship, we can differentiate the expression for electric potential:

Calculating the Gradient

Starting with the potential:

V = k * Q / r

Taking the derivative with respect to r:

dV/dr = -k * Q / r²

Connecting Electric Field and Potential

Substituting this result into the equation for the electric field gives:

E = -(-k * Q / r²) = k * Q / r²

This confirms that the electric field due to a point charge is indeed the negative gradient of the electric potential.

Summary

In summary, the electric field and electric potential are interconnected through the relationship:

E = -dV/dr

This means that the electric field is the negative rate of change of electric potential with respect to distance, illustrating how these two concepts are fundamentally linked in electrostatics.