Finding the magnitude and direction of current in resistors in a given circuit

Question

Considering the circuit shown in the image attached:

https://ibb.co/wLTfxY3

a) Can the circuit be reduced to a single resistor connected to the batteries?

b) And what will be the magnitude of the current and its direction in each resistor?

Answer 1

As no numerical values for the Rs and Es are given, I can only describe the general procedure.

You need to write two Kirchoff loop equations and one Kirchoff junction equation as follows. We need to assume that the Rs and Es are known.

Loop1; E1 -I1R1 + I2R2 -E2 =0

Loop2: E2 -I2R2 -I3R3 = 0

Junction a: I1 + I2 = I3

Substitute I3 from junction a equation for I3 in Loop2.

Simplify to get E2 - I2(R3 + R2) - I1R3 = 0

This last equation and the equation for Loop1 have two unknowns, I1 and I2 that can be solved simultaneously for I1 and I2.

Once I1 and I2 are calculated, they can be put into the junction a equation to get I3.

I didn't see a way to get only one resistor.

Answer 2

Please note a slight mistake in my expression for R_eq. The correction expression for R_eq is as below:

R_eq = R1|| R2 || R3 = (R₁R₂R₃) / (R₁R₂ + R₂R₃ + R₃R₁)