If cell constant doubles and resistance of cell is halved, what will be new resitivity?

resistivity becomes 1/4 times

The correct answer is option 4. resistivity becomes 1/4 times.

To determine the effect on resistivity when the cell constant doubles and the resistance of the cell is halved, we need to understand the relationship between resistivity, cell constant, and resistance.

Cell Constant (\(K\)) is defined as the ratio of the distance between the electrodes to the area of the electrodes.

\( K = \frac{d}{A} \)

Resistance of the Cell (\(R\)) is measured in the cell depends on the resistivity (\(\rho\)), cell constant (\(K\)), and the cell's dimensions.

The relationship is given by:

\(R = \rho \cdot \frac{K}{A}\)

Rearranged to:

\(R = \rho \cdot K\)

Resistivity is a material property and does not directly change with cell dimensions or cell constant, but can be calculated from the resistance and cell constant.

Given Situation:

If cell constant doubles:

New cell constant \( K' = 2K \)

If resistance of the Cell is halved:

New resistance \( R' = \frac{R}{2} \)

Finding the New Resistivity:

Using the formula \( R = \rho \cdot K \), we can express the resistivity in terms of the old and new conditions:

Original Resistivity: \(\rho = \frac{R}{K}\)

New Resistivity:  With new resistance \( R' = \frac{R}{2} \) and new cell constant \( K' = 2K \), we use:

\(R' = \rho' \cdot K'\)

\(\frac{R}{2} = \rho' \cdot (2K)\)

Solving for \( \rho' \):

\(\rho' = \frac{\frac{R}{2}}{2K}\)

\(\rho' = \frac{R}{4K}\)

\(\rho' = \frac{\rho}{4}\)

The new resistivity (\(\rho'\)) is one-fourth of the original resistivity (\(\rho\)). Therefore, the correct answer is: 2. Resistivity becomes \(\frac{1}{4}\) times.