The potentiometer wire AB is made of two parts AN and NB of cross sectional radii 2r and r respectively. Hence, the cross sectional area of the parts AN and NB are in the ratio 1 :4 and the resistance per unit length will be in the ratio 4:1.Therefore the potential gradient in the part NB will be four times that in the part AN.i.e. \\\\(V_{NB} = 4 V_{AN}\\\\)But, \\\\(V_{AN} = 1 V\\\\)\\\\(V_{NB} = 4 V\\\\)Let, \\\\(CN = x m\\\\) From above discussion,\\\\(ND = (0.2-x) m\\\\)Now, \\\\(V_{CD} = V_{CN}x + (0.2-x)\ imes 4\\\\)\\\\(0.5 = (1)x + (0.8-4x)\\\\)\\\\((0.3)x = 0.3\\\\)\\\\(\ herefore x = 0.1m\\\\) But, \\\\(CN = \\dfrac {t}{3}\\\\) i.e. \\\\(x = \\dfrac {t}{3}\\\\)Hence, \\\\( 0.1m = (\\dfrac {t}{3})cm\\\\)\\\\(\ herefore t = 0.3 m = 30cm\\\\)