Question

A wire of uniform cross-section and length l has a resistance of 4 ohms. The wire is cut into 4 equal pieces. Each piece is then stretched to length l. Thereafter, the four wires are joined in parallel. Calculate net resistance.

Answer 1

Let the resistivity of the material be $\rho$, length of the wire be $l$ and the area of cross-section be $A$.

Resistance of the wire = $R=\rho \frac{l}{A}=4\Omega$

When the wire is cut into four equal pieces and stretched to the initial length $l$, the new length of the wire will be the same for each piece but the area will get reduced to one-fourth.

Therefore,

$R^{\prime }=\rho \frac{l^{\prime }}{A^{\prime }}=\rho \frac{l}{A/4}=4\rho \frac{l}{A}=4R=16\Omega$

When n resistances of Resistance R are connected in parallel the resultant resistance = R/n.
Therefore when 4 resistances of resistance R' are connected in parallel then the equivalent resistance is R'/4.
Therefore net resistance = $R_{net}=\frac{16}{4}=4\Omega$.