Derive the expression for equivalent resistance of series and parallel combination of resistances. Find the ratio of the equivalent resistances of four wires of equal resistances first connected in series then connected in parallel

Open in App

Solution

$I f t h r e e r e s i s \tan c e s a r e c o n n e c t e d i n s e r i e s a n d c u r r e n t p a s s e s t h r o u g h t h e m i s I . S o t h i s c u r r e n t I w i l l p a s s t h r o u g h t h e r e s i s \tan c e R 1, R 2 a n d R 3 . A p p l y i n g O h m ’ s l a w, i t c a n b e f o u n d t h a t v o l t a g e d r o p s a c r o s s t h e r e s i s \tan c e s w i l l b e V_{1} = I R_{1}, V_{2} = I R_{2} a n d V_{3} = I R_{3} . N o w, i f t o t a l v o l t a g e a p p l i e d a c r o s s t h e c o m b i n a t i o n o f r e s i s \tan c e s i n s e r i e s, i s V . V = I R_{1} + I R_{2} + I R_{3} \dots \dots \dots \dots . (1) V = I R R = R_{1} + R_{2} + R_{3} W h e n t h r e e r e s i s \tan c e s a r e c o n n e c t e d i n p a r a l l e l . I f e l e c t r i c p o t e n t i a l d i f f e r e n c e i s a p p l i e d a c r o s s t h i s c o m b i n a t i o n, t h e n i t w i l l d r a w a c u r r e n t I (s a y) . A s t h i s c u r r e n t w i l l g e t t h r e e p a r a l l e l p a t h s t h r o u g h t h e s e t h r e e e l e c t r i c a l r e s i s \tan c e s, t h e c u r r e n t w i l l b e d i v i d e d i n t o t h r e e p a r t s . S a y c u r r e n t s I_{1}, I_{1} a n d I_{1} p a s s t h r o u g h r e s i s t o r R_{1}, R_{2} a n d R_{3} r e s p e c t i v e l y . W h e r e t o t a l s o u r c e c u r r e n t I = I_{1} + I_{2} + I_{3} p u t t i n g, I = V / R \frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} W h e n 4 r e s i s \tan c e a r e c o n n e c t e d i n s e r i e s R_{s} = 4 R a n d i f c o n n e c t e d i n p a r a l l e l R_{P} = R / 4 \frac{R_{S}}{R_{p}} = \frac{16}{1}$

Suggest Corrections

Similar questions

n

n

[

]

ρ_{1}

ρ_{2}

Join BYJU'S Learning Program