Ten identical cells connected in series are needed to heat a wire of length one meter and radius ‘r’ by ${10}^{o}C$ in time ‘t’. How many cells will be required to heat the wire of length two meter of the same radius by the same temperature in time ‘t’ ?

detailed solution

Correct option is A

Let ρ be resistivity of the material of the wire and r be radius of the wire. Therefore, resistance of 1 m wire isR=ρ(1)πr2=ρπr2  ∵R=ρlALet ε be emf of each cell. In first case,10 cells each of emf ε are connected in series to heat the wire of length 1 m by ΔT(=10oC)in time t.∴  (10ε)2Rt=msΔT                      ........(i)In second case, Resistance of same wire of length 2 m isR'=ρ(2)πr2=2ρπr2=2RLet n cells each of emt ε are connected in series to heat the same wire of length 2 m, in the same temperature ΔT(=10oC) in the same time t∴  (nε)2t2R=(2m)sΔT                     .....(ii)Divide (ii) by (i), we getn2200=2⇒n2=400∴  n=20