Two cells with electromotive forces E1 and E2 and internal resistances r1 and r2 are connected to a resistor R as shown.

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Question 38

Two cells with electromotive forces E1 and E2 and internal resistances r1 and r2 are connected to a resistor R as shown.

The terminal potential difference across cell 1 is zero.

Which expression gives the resistance of resistor R?

Reference: Past Exam Paper – November 2019 Paper 12 Q34

Solution:

Answer: A.

Let the current in the circuit be I.

From Kirchhoff’s law,

Sum of e.m.f. in the circuit = Sum of p.d. in the circuit

E₁ + E₂= I (r₁ + r₂ + R)

R + r₁ + r₂= (E₁ + E₂) / I                     eqn (1)

However, none of the options available contains the current I. So, we need to substitute I by other quantities.

Consider the cell 1.

e.m.f. of cell 1 = terminal p.d. + lost volts

It is given that the terminal p.d. is zero. So,

e.m.f. of cell 1 = lost volts

E₁ = I × r₁

Current I = E₁ / r₁

Replace the current I in equation (1),

R + r₁ + r₂= (E₁ + E₂) × r₁ / E₁

R + r₁ + r₂= E₁r₁/E₁ + E₂r₁/E₁

R + r₁ + r₂= r₁ + E₂r₁/E₁

R = r₁ + E₂r₁/E₁– r₁ – r₂

R = E₂r₁/E₁– r₂

Note that r₂can be written as E₁r₂/E₁ so as to obtain the same denominator.

R = E₂r₁/E₁– E₁r₂/E₁

R = (E₂r₁–1r₂) / E₁