Two resistances R 1 = X Ω and R 2 = 1 Ω are connected to a wire A B of uniform resistivity, as shown in the figure. The radius of the wire varies linearly along its axis from 0 . 2 mm at A to 1 mm at B . A galvanometer G connected to the center of the wire, 50 cm from each end along its axis, shows zero deflection when A and B are connected to a battery. The value of X is _______.

Step-by-step solution

Resistance of frustum shaped conductor shown is R = ρ I π a b For the shown conductor in the diagram. r = a + b 2 = 0 . 2 + 1 2 = 0 . 6   mm Thus, the resistance of left half is P = ρ × 0 . 5 × 10 6 π × 0 . 2 × 0 . 6 And the resistance of right half is Q = ρ × 0 . 5 × 10 6 π × 0 . 6 × 1 For Wheatstone to be balanced R 1 P = R 2 Q ⇒ X π × 0 . 2 × 0 . 6 ρ × 0 . 5 × 10 6 = 1 π × 0 . 6 × 1 &#961