Question

In Figure the resistances are R1 = 1.0 Ohm  and R2 = 2.0 Ohm, and the ideal batteries have emfs

E1=  2.0 V and
E2 =E3=  4.0 V.
What are the currents in battery 1, and in battery 2?

Answer 1

Calculate the currents through the batteries in the following circuit:

Apply Kirchhoff's voltage law for loops E1>R1>R2>E2 and E1>R1>R1>E3>R1>R1, as well as current law assuming that currents flow in the direction of the emf:

$$E_1-E_2=2I_1R_1-I_2R_2,\\ E_1-E_3=2I_1R_1-I_3R_1,\\ I_1+I_2+I_3=0.$$

We have three unknown currents, three equations. The solution of the system is

$$I_1=-0.75\text{ A},\\ I_2=0.25\text{ A},\\ I_3=0.5\text{ A}.$$

As we see, the first current flows in the opposite direction regarding the direction of EMF from E1.