A wire of uniform resistance / m is bent into a circle of radius r and another piece of wire with length 2 r is connected between points A and B(A O B) as shown in figure. The equivalent resistance between points A and B is _ _ _ _ .

Step-by-step solution

The wire bent into a circle of radius r has resistance r for the semicircular arcs, and the straight wire of length 2r has resistance 2r . The three paths between A and B (two semicircular arcs of the circle and one straight wire through center) are in parallel. 1 R_ AB = 2 r + 1 2r + 2 3 r = 1 r [ 2 + 1 2 + 2 3 ] = 1 r ( 12 + 3 + 4 6 ) = 1 r 16 + 3 6 R_ AB = r ( 6 16 + 3 ) = 6 r 3 + 16