A uniform wire of resistance \( 36\,\Omega \) is cut into a number of equal pieces, and all the pieces are then connected in parallel. If the equivalent resistance of the combination is \( 1\,\Omega \), the number of pieces is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 6
If the wire is cut into \( n \) equal pieces, each piece has \( 1/n \) of the length and so \( 1/n \) of the resistance. When \( n \) equal resistances are joined in parallel, the equivalent resistance is one of them divided by \( n \).
For a wire of resistance \( R \):
\( \displaystyle R_{\mathrm{piece}}=\frac{R}{n} \)
\( \begin{aligned} R_{\mathrm{eq}} &= \frac{R_{\mathrm{piece}}}{n} \\ &= \frac{R}{n^{2}} \end{aligned} \)
Putting \( R=36\,\Omega \) and \( R_{\mathrm{eq}}=1\,\Omega \):
\( \displaystyle 1\,\Omega=\frac{36\,\Omega}{n^{2}} \)
\( n^{2}=36 \)
\( \Rightarrow\; n=6 \)
Each piece then has a resistance of \( 6\,\Omega \), and six such pieces in parallel give \( 1\,\Omega \).