Question

Derive an expression for the effective resistance of three resistances in series.

Answer 1

Consider three resistors in series as shown, with resistances R${}_1$, R${}_2$ and R${}_3$ respectively

If V${}_1$, V${}_2$ and V${}_3$ are potential differences across them respectively, the total potential difference across the system of resistors,

V = V${}_1$ + V${}_2$ + V${}_3$

ie.,V = $I$R${}_1$ + $I$R${}_2$ + $I$R${}_3$ = $I$(R${}_1$ + R${}_2$ + R${}_3$) $\to$(1)

(Since the current through them is the same)

Let the three resistors be replaced by a single resistor of resistance R_S called the equivalent or effective resistor.

Hence, for equivalent resistor, V = $I$R${}_S$$\to$ (2)

From (1) and (2),

We have,$R_s=R_1+R_2+R_3$