Question

Explain and derive equivalent Resistance formula for 3 Resistors connected in Pure series. Generalize the formula?

Answer 1

When three resistors are concealed in derives then the same current passes through each resistor but the voltage drop is different for each resistor.

Complete step by step solution:
According to Ohm’s formula V = IR.
When three resistors are connected in series then the same current passes through each resistor but the voltage drop is different for each resistor. If V is the applied voltage and ${V_1},\,\,\,{V_2}$ and ${V_3}$ is voltage across resistance ${R_1},\,\,{R_2}\,\,and\,\,{R_3}$ respectively then.
$V = \,\,{V_1}\,\, + \,\,{V_2}\,\, + \,\,{V_3}$ …(1)

According to ohm’s law
V = IR
$\therefore$ $I{R_{eq}}\,\, = \,\,I{R_1} + I{R_2} + I{R_3}$
$I{R_{eq}}\,\, = \,\,I\left( {{R_1} + {R_2} + {R_3}} \right)$
${R_{eq}}\,\, = \,\,{R_1} + {R_2} + {R_3}$

So, the equivalent resistance or the total resistance of the circuit can be defined as a single value of resistance that can replace any number of resistors connected in series without altering the value of the current or the voltage of the circuit.
If we have n resistance connected in series, then generalized formula for equivalent resistance is
${R_{eq}}\,\, = \,\,{R_1} + {R_2} + {R_3}....... + {R_n}$

We know that applied voltage is (v)
$\therefore \,\,V = {V_1} + {V_2} + {V_3}$
We also know that
V = IR (from ohm’s law)
$\therefore \,I\,{R_{eq}} = I{R_1} + I{R_2} + I{R_1}$
$\Rightarrow I\,{R_{eq}} = I\left( {{R_1} + {R_2} + {R_3}} \right)$
$\Rightarrow {R_1} + {R_2} + {R_3}$
So, the equivalent resistance or the total resistance of the circuit can be defined as a single value of resistor connected in series with altering the values of the current or voltage in the circuit.

Note: If three or more resistors with the same value are connected in parallel, then the equivalent resistance is equal to $\dfrac{R}{n}$.