Question

Suppose there are three resistors A, B, and C having resistances r₁, r_2, and r₃ respectively. If R represents their equivalent resistance, establish the following relation R = r₁ + r₂ + r₃ when joined in series.

Answer 1

Let the three resistors be joined in series in given figure.

Let potential at P, Q, R and S are V_p, V_Q, V_R, and V_S respectively and current i flows in the circuit.

Applying Ohm's law,

V_p + V_Q = ir₁     ...(i)

V_Q - V_R = ir₂      ....(ii)

and V_R - V_S = ir₃   ....(iii)

If the total effective resistance between P and S be R', then the potential difference:

V_P - V_R = iR'    ....(A)

Adding (i), (ii) and (iii)

V_P - V_Q + V_Q - V_R + V_R - V_S = ir₁ + ir₂ + ir₃

or V_P - V_S = i (r₁ + r₂ + r₃)    .... (B)

Comparing (A) and (B), we get

iR' = i (r₁ + r₂ + r₃)

or    R' = r₁ + r₂ + r₃