The EMF of the battery is 20 V. The efficiency of the battery is 0.8 at a current of 4A. What is the internal resistance of the battery?

The efficiency of the current source is:
η = P / Ptotal = R / (R + r)
According to Ohm’s law for a closed circuit with EMF, the current strength is determined by the expression:
I = Ε / (R + r), where U is the voltage in the circuit, R is the resistance in the circuit, r is the internal resistance of the source.
Let us express r:
r = E / I-R
For uchatka chains:
I = U / R
Let us express R from this expression:
R = U / I
Let’s equate the current strength:
Ε / (R + r) = U / R, divide both sides by E and multiply by R:
R / (R + r) = U / E
Hence U = E * η
R = U / I = E * η / I
Let’s plug all this into the expression for r:
r = E / I-R = E / I-E * η / I = 20 / 4-20 * 0.8 / 4 = 1 Ohm.
Answer: The internal resistance of the battery is 1 ohm.