### Long Question Solution

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#### Kinematics

Introduction to Kinematics

# Electrostatics - Long Question Answer 7

Question: Use Coulomb’s law to define potential difference between two points near a static charge. Also derive an expression for it.
Ans: The electric potential difference between two points inside the electric field is defined as the amount of work done in moving a unit positive charge (test charge) from one point to another point against the field. If WBA is the amount of work done in moving unit positive charge from point B to point A. then, it is equal to the potential difference between two points A and B. i.e. VAB = WBA
Suppose a unit positive is at any point A so that OM = x from the point charge +q at O. we have to find electric potential difference between two points A and B. Since the force experienced by unit positive charge is equal to the electric intensity at that point, then,
F = 1/4πϵ0 (q/x2),

The work done in moving a unit positive charge from M to N through small-displacement dx is,
dW = F * (-dx) = 1/4πϵ0 (q/x2) dx

Therefore, the total work done in moving a unit positive charge from point B to point A is
WBA = ∫dW
= ∫r1r2 1/4πϵ0 (q/x2) dx
= q/4πϵ0 (1/r2 - 1/r1) = VAB

Hence, the potential difference between two points A and B is
VAB = q/4πϵ0 (1/r2 - 1/r1)

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