A circular ring of radius r is uniformly charged with a linear charge density . Determine the electric potential at a point on the axis located at a distance x from the centre of the ring. Subsequently, use this potential expression to determine the electric field at the same point.

Step-by-step solution

Consider an elemental charge dq on the ring of radius r . The linear charge density is , so dq = dl . The total charge on the ring is Q = dl = (2 r) . The electric potential V at a point on the axis at a distance x from the centre is given by integrating the potential due to each elemental charge: V = 1 4 ₀ dq r^2 + x^2 Since the distance r^2 + x^2 is constant for all points on the ring: V = 1 4 ₀ r^2 + x^2 dq = Q 4 ₀ r^2 + x^2 Substituting Q = 2 r : V = 2 r 4 ₀ r^2 + x^2 = r 2 ₀ (r^2 + x^2)^ 1/2 The electric field