A rod of length l having resistance R, is cut into two equal parts. These parts are connected in parallel then new resistance shall be

The correct option is R/4.

Step-by-step Explanation:

The resistance of a uniform conductor (like a metal rod) is directly proportional to its length ($l$) and inversely proportional to its cross-sectional area ($A$). This relationship is mathematically expressed as:

$R=\rho \frac{l}{A}$

where $\rho$ represents the electrical resistivity of the material of the rod.

Step 1: Find the resistance of each piece after cutting
When the rod of length $l$ is cut into two equal halves, the length of each half becomes $l^{\prime }=\frac{l}{2}$. The material's resistivity ($\rho$) and the cross-sectional area ($A$) remain the same.

Let $R^{\prime }$ be the resistance of each of the two halves. We calculate it as:

$R^{\prime }=\rho \frac{l^{\prime }}{A}=\rho \frac{l/2}{A}=\frac{1}{2}\left(\rho \frac{l}{A}\right)=\frac{R}{2}$

Thus, each half has a resistance of $\frac{R}{2}$.

Step 2: Calculate the equivalent resistance of the parallel connection
When these two halves, each with resistance $R^{\prime }=\frac{R}{2}$, are connected in parallel, the new equivalent resistance $R_{\text{new}}$ is given by the formula:

$\frac{1}{R_{\text{new}}}=\frac{1}{R^{\prime }}+\frac{1}{R^{\prime }}=\frac{2}{R^{\prime }}$

Taking the reciprocal of both sides gives:

$R_{\text{new}}=\frac{R^{\prime }}{2}$

Now, substitute the value of $R^{\prime }=\frac{R}{2}$ into this expression:

$R_{\text{new}}=\frac{R/2}{2}=\frac{R}{4}$

Therefore, the new equivalent resistance of the parallel combination is R/4.