Deduce. ohms law from elementary. Ideas and hence write an expression for resistance and resistivity

Open in App

Solution

$T h e e l e m e n t a r y i d e a o f o h m' s l a w s t a t e s t h a t p o t e n t i a l d i f f e r e n c e V a c r o s s a c o n d u c t o r i s d i r e c t l y p r o p o r t i o n a l t o t h e c u r r e n t I p a s \sin g t h r o u g h t h e c o n d u t o r w h e n p h y s i c a l c o n d i t i o n s o f t h e c o n d u c t o r a r e n o t c h a n g e d . i . e . V α I o r \frac{V}{I} = c o n s \tan t (R) W h e r e R i s t h e c o n s \tan t k n o w n a r r e s i s \tan c e o f t h e c o n d u c t o r . R e s i s \tan c e i s d i r e c t l y p r o p o r t i o n a l t o t h e l e n g t h o f t h e c o n d u c t o r a n d i n v e r s e l y p r o p o r t i o n a l t o t h e a r e a o f c r o s s e c t i o n o f t h e c o n d u c t o r . T h e r e f o r e R α \frac{l}{A}, w h e r e R, l a n d A a r e r e s i s \tan c e, l e n g t h a n d a r e a o f c r o s s e c t i o n r e s p e c t i v e l y . O n r e m o v i n g t h e s i g n o f p r o p o r t i o n a l i t y, w e g e t : R = \frac{ρ l}{A}, w h e r e ρ i s t h e r e s i s t i v i t y o f t h e c o n d u c t o r o f r e s i s \tan c e R w h i c h d e p e n d s o n t h e n a t u r e o f t h e c o n d u c t o r . N o w I = n e A v_{d}, w h e r e n i s t h e n u m b e r o f e l e c t r o n s p r e s e n t i n u n i t v o l u m e o f t h e c o n d u c t o r d r i f t i n g w i t h v e l o c i t y v_{d} . B u t v_{d} = \frac{e E}{m} τ, s o p u t t i n g t h e v a l u e o f v_{d} i n t h e a b o v e e q n o f c u r r e n t, w e g e t, I = n e A \times \frac{e E}{m} τ f u r t h e r w e k n o w t h a t E = \frac{V}{l}, s o I = n e A \times \frac{e V}{m l} τ A g a i n \frac{V}{I} = R = \frac{m l}{n e^{2} A τ} a n d ρ = \frac{R A}{l} = \frac{m}{n e^{2} τ} T h e s e a r e t h e r e q u i r e d e x p r e s s i o n s .$

Suggest Corrections

Similar questions

Q. Deduce the expression for the the equivalent resistance of three resistors connected in Parallel.

Q. Deduce the expression for the the equivalent resistance of three resistors connected in series.

State Biot – Savart law. Deduce the expression for the magnetic field at a point on the axis of a current carrying circular loop of radius ‘R’ at a distance ‘x’ from the centre. Hence, write the magnetic field at the centre of a loop.

Q. State Kirchoff's law for an electrical network. Using these laws deduce the condition for balance in a wheatstone bridge.
Three resistors

2 Ω, 4 Ω

and

5 Ω

are combined in parallel. What is the total resistance of the combination?

Q. State Kirchhoff's law for an electrical network. Using these laws deduce the condition for balance in a Wheatstone bridge.
A wire of resistance 4R is bent in the form of a circle. What is the effective resistance between the ends of the diameter?

Join BYJU'S Learning Program