Two cells of emfs `epsilon_(1)and epsilon_(2)` and internal resistances `r_(1)and r_(2)` respectively are connected in parallel . Obtain expressions for the equivalent (i) resistance and , (ii) emf of the combination.

Explanation

According to Kirchhoff first law ,

`I=I_(1)+I_(2)` . . . (i)
Potential across upper branch `V=epsilon_(1)-I_(1)r_(1)` . . . (ii)
Potential across lower branch `V=epsilon_(2)-I_(2)r_(2)` . . . (iii)
Potential across equivalent branch `V=epsilon_(eq)-I_("req")` . . . (iv)
Substituting the values of I , `I_(1)andI_(2)` from equations (ii) , (iii) and (iv) in equation (i),
`{:((epsilon_(eq))/(r_(eq))-(V)/(r_(eq))=(epsilon_(1))/(r_(1))-(V)/(r_(1))+(epsilon_(2))/(r^(2))-(V)/(r_(2))," ",(epsilon_(eq))/(r_(eq))-(V)/(r_(eq))=(epsilon_(1))/(r_(1))+(epsilon_(2))/(r_(2))+((1)/(r_(eq))+(1)/(r_(eq)))V):}`
Comparing both the sides , `(epsilon_(eq))/(r_(eq))=(epsilon_(1))/(r_(1))+(epsilon_(2))/(r_(2))and(1)/(r_(eq))=(1)/(r_(1))+(1)/(r_(2))`