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Resistivity

Two cubes A a...

Question

Two cubes A and B are of the same material .the side of B is thrice as that of A. Find the ratio of R_A/R_B?

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Solution

$L e t ρ b e t h e r e s i s t i v i t y o f t h e m a t e r i a l . a i s t h e e d g e l e n g t h o f t h e c u b e A, 3 a i s t h e e d g e o f t h e c u b e B . W h e n w e c o n n e c t t h e c u b e a c r o s s t h e t w o a d j e c e n t s u r a f a c e s, t h e n l e n g t h o f t h e c o n d u c t o r (c u b e A) i s a a n d a r e a o f c r o s s - s e c t i o n i s a^{2} . L e n g t h o f t h e c o n d u c t o r B i s 3 a a n d a r e a o f c r o s s - s e c t i o n i s 9 a^{2} . R_{A} = \frac{ρ l e n g t h}{a r e a} = \frac{ρ a}{a^{2}} = \frac{ρ}{a} s i m i l a r l y R_{B} = \frac{ρ}{3 a} \frac{R_{A}}{R_{B}} = \frac{\frac{ρ}{a}}{\frac{ρ}{3 a}} = 3 : 1$

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