Question

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

Answer 1

Hint: 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 \[\dfrac{R}{n}\].