Question

A metal wire of resistance $R$ is cut into three equal pieces that are then connected side by side to form a new wire, the length of which is equal to one third of the original length. The resistance of this new wire is: -
A. $R$
B. $3R$
C. $\dfrac{R}{9}$
D. $\dfrac{R}{3}$

Answer 1

Resistance of a wire is directly proportional to the length of the wire (so if a wire is cut into half the resistance of each of the new wires becomes half of the resistance of the original wire). As in this question the wire is cut into three equal pieces, each of the new wire will have a length equal to one-third of the original length and the resistance of those wires becomes one-third of the original.

Formulas used:
$R = \dfrac{{\rho l}}{A}$ $\left( {R \propto l} \right)$
$\Rightarrow \dfrac{1}{{{R_p}}} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + \dfrac{1}{{{R_3}}}$

Complete step by step answer:
Resistance is a measure of the opposition to the flow of the current in an electrical circuit.The resistance of a wire is directly proportional to the length of the wire and inversely proportional to the area of the wire thus resistance becomes,
$R = \dfrac{{\rho l}}{A}$
where $l$ is the length of the wire, $A$ is the cross-sectional area of the wire and $\rho$ is the resistivity of the wire.

From the above formula we can see that $\left( {R \propto l} \right)$ as the rest of the variables( $A$ and $\rho$) are constant. Since all the wires are of the same length.The resistance of the new wires is:
${R_1} = {R_2} = {R_3} = \dfrac{R}{3}$.......(Where $R$ is the resistance of the original wire)
When the pieces are joined side by side to each other and the length of the new combination becomes one third of the original length we are sure that they are now in a parallel combination, so the resistance the new wire formed by joining them will equal to,
$\dfrac{1}{{{R_n}}} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + \dfrac{1}{{{R_3}}}$......(${R_n}$ is the resistance of the new circuit)

Now by substituting the values,
${R_1} = {R_2} = {R_3} = \dfrac{R}{3}$
$\Rightarrow \dfrac{1}{{{R_n}}} = \dfrac{1}{{\dfrac{R}{3}}} + \dfrac{1}{{\dfrac{R}{3}}} + \dfrac{1}{{\dfrac{R}{3}}}$
$\Rightarrow \dfrac{1}{{{R_n}}} = \dfrac{3}{R} + \dfrac{3}{R} + \dfrac{3}{R}$
$\Rightarrow \dfrac{1}{{{R_n}}} = \dfrac{9}{R}$
$\therefore {R_n} = \dfrac{R}{9}$

So, the correct answer is option C.

Note: If the resistance is connected in series, then the total value always increases if it’s in parallel the total value decreases. This can be helpful to eliminate options if the question is in multiple choice type. Here as our new wires are joined in a parallel connection (as original length becomes one third) so the total resistance of the new connection decreases from the original value.