Ten identical cells connected in series are needed to heat a wire of length one meter and radius ‘r’ by 10oC 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’ ?

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