Question

A piece of wire having a resistance R is cut into five equal parts.

1. How will the resistance of each part of the wire change compared with the original resistance?
2. If the five parts of the wire are placed in parallel, how will the resistance of the combination change?
3. What will be ratio of the effective resistance in series connection to that of the parallel connection?

Answer 1

1. Consider a piece of wire having resistance R. It was cut into 5 equal parts. So number of equal resistors is 5. n = 5
   When ‘n’ resistors of equal resistance R are connected in series, the equivalent resistance is ‘nR’
   R_S = nR
   ⇒ R_S = 5R
   ⇒ R = `R_s/5`
   ⇒ R = 0.2 R_S
   Each part of resistance ‘R’ is equal to 0.2 times the original resistance.
2. Effective Resistance of 5 Resistors
   `1/R_P = 1/R + 1/R + 1/R + 1/R + 1/R = 5/R`
   `R_P = R/5` = 0.2 R 
3. Effective resistance of series combination R_S = 5R
   Effective resistance of parallel combination
   `R_P = R/5`
   The ratio of series connection to the parallel connection
   `R_S/R_P = (5 R)/(R//S)`
   (2) (9 - R₂ - + R₂) = (9 - R₂)R₂
   `R_S/R_P = 5 R xx 5/R = 25`
   R_S : R_P = 25 : 1