A piece of wire of resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is R', then the ratio RR′\dfrac{\text{R}}{\text{R}'}R′R is ...............
125\dfrac{1}{25}251
15\dfrac{1}{5}51
5
25
Reason — Given,
Wire of resistance R is cut into five equal parts hence, resistance of each part is R5\dfrac{\text{R}}{5}5R
1R′=1R5+1R5+1R5+1R5+1R5=5R+5R+5R+5R+5R=25R⇒R′=R25\dfrac{1}{\text{R}'} = \dfrac{1}{\dfrac{\text{R}}{5}} + \dfrac{1}{\dfrac{\text{R}}{5}}+ \dfrac{1}{\dfrac{\text{R}}{5}} + \dfrac{1}{\dfrac{\text{R}}{5}} + \dfrac{1}{\dfrac{\text{R}}{5}} \\[1em] = \dfrac{5}{\text{R}} + \dfrac{5}{\text{R}} + \dfrac{5}{\text{R}} + \dfrac{5}{\text{R}} + \dfrac{5}{\text{R}} \\[1em] = \dfrac{25}{\text{R}} \\[1em] \Rightarrow \text{R}' = \dfrac{\text{R}}{25}R′1=5R1+5R1+5R1+5R1+5R1=R5+R5+R5+R5+R5=R25⇒R′=25R
Hence,
RR′=RR25⇒RR′=25\dfrac{\text{R}}{\text{R}'} = \dfrac{\text{R}}{\dfrac{\text{R}}{25}} \\[1em] \Rightarrow \dfrac{\text{R}}{\text{R}'} = 25R′R=25RR⇒R′R=25
