A uniformly charged non-conducting wire with linear charge density is bent into a closed planar loop. The loop consists of two concentric circular arcs of radii R and eR (where e is the base of the natural logarithm), both subtending an angle of 90^ at their common centre, and two straight radial wire segments connecting the ends of the arcs. The net electric potential at the common centre of the arcs is:

Step-by-step solution

The loop consists of four parts: two circular arcs and two straight radial segments. The electric potential at the centre due to a circular arc subtending an angle is V = 4 ₀ . For each 90^ ( /2 radians) arc, the potential is: V_ arc = ( /2) 4 ₀ = 8 ₀ Total potential from the two arcs is 2 8 ₀ = 4 ₀ . For a straight radial segment, the potential at the centre is found by integrating the contributions of infinitesimal elements dr at distance r : V_ straight = _ R ^ eR 1 4 ₀ dr r = 4 ₀ [ r]_ R ^ eR = 4 ₀ ( eR R ) = 4