Line Integral of Electric Field

Description:

The multiplication of electric field and the path length gives Line integral of electric Field.

Let there be two point A and B on a non uniform electric field (magnitude and direction at every point is different). The irregular line denotes the path travelled by charge. With changing Electric field, the path travelled by a charge also changes at every point on the curve.

To find line integral at every point, take a very small length dl on the path. So that dl makes a certain angle θ with the electric field. On resolving dl into two components as dlcosθ and dlsinθ the component

of dl along E is dl cosθ. As shown in figure.

Potential difference dV = Edr.

So line integral to move charge from A to B = A∫B EdrCosθ

V_A - V_B = A∫B E→.dr→

To move charge from B to A (V_A - V_B) = - B∫A E→.dr→

Note−

in a variable electric field, potential difference between two point is equal to negative of line integral of electric field.