In the circuit shown in fig `E_1 = 3 volts, E_2 = 2volts, E_3 = 1volt and R = r_1 = r_2= r_3= 1 ohm.`

(i) Find the potential difference between the points A and B and the currents through each branch.
(ii) If `r_2` is short circuited and the point A is connected to point B, find the current through `E_1, E_2, E_3` and the resistor R.

Answer

A, B

Explanation

(i) `V_(AB) = (SigmaE//r)/(Sigma 1/r) = (E_1/r_1 + E_2/r_2 + E_3/r_3)/(1/r_1 + 1/r_2 + 1/r_3)`
(ii)
Applying Kirchoff's law in PQRUP starting from P moving
clockwise
`I_1r_1 - E_1 + E_2 = 0 or I_1-3+2 = 0`
or `I_1 = 1 amp`
Applying Kirchoff's law in URSTU starting from U moving
clockwise
`-E_2 + E_3 - I_3r_3 = 0 `
or `-2 + 1-I_3 = 0`
or `I_3 = -1 amp`
`(I_1+I_2+ I_3) R -E_2 = 0 or (1+ I_2-1) R=2`
or `I_2 = 2amp`
Current through `R is I_1+ I_2 + I_3 = 2A`.