Question

Suppose two batteries, with unequal emfs of 2.00 V and 3.00 V, are connected as shown in (Figure 1) If each internal resistance is r = 0.400? , and R = 4.00 ?, what is the voltage across the resistor R?

Answer 1

Call the emf of the upper battery V1
Call the emf of the lower battery V2

Current through V1 (positive if to the right): I1
Current through V2 (positive if to the right): I2
Current through R (positive if to the left): I3

Current branch law equation:
I1 + I2 = I3

Voltage loop law about the loop containing V1, r, and R:
V1 - I1*r - I3*R = 0

Voltage loop law about the loop containing V2, r, and R:
V2 - I2*r - I3*R = 0

Solve three equations for three unknowns (I1, I2, and I3):
Algebra not displayed, results below:
I1 = ((R+r)*V1 - R*V2)/(r*(2*R+r))
I2 = ((R+r)*V2 - R*V1)/(r*(2*R+r))
I3 = (V1+V2)/(2*R+r)

To find the voltage across external resistor R, use the result for I3:
V_out = I3*R

Resulting expression:
V_out = (V1+V2)*R/(2*R+r)

Data:
V1:=2 Volts; V2:=3 Volts; R:=4.0 Ohms; r:=0.40 Ohms;

V_out = (V1+V2)*R/(2*R+r) = (3+2)*4/(2*4+0.4)= 20/8.4=2.38.

please find new solutions thnx