QJEEM25Ph302f: JEE Main 2025 Physics - Class 12 JEEM25Ph302f Solution: A parallel plate capacitor was made with two rectangular…

The capacitance of a parallel plate capacitor is given by

$C = \frac{\epsilon_0 A}{d},$

where:

$\epsilon_0$ is the permittivity of free space,

$A$ is the area of a plate, and

$d$ is the separation between the plates.

A change in the capacitor’s dimensions will change its capacitance by a factor of

$\frac{C_{\text{new}}}{C_{\text{initial}}} = \frac{A_{\text{new}}}{A_{\text{initial}}} \cdot \frac{d_{\text{initial}}}{d_{\text{new}}}.$

The initial dimensions are:

Length: $l = 3\,\text{cm} = 0.03\,\text{m},$

Breadth: $b = 1\,\text{cm} = 0.01\,\text{m},$

Hence, the original area is

$A_{\text{initial}} = 0.03 \times 0.01 = 3 \times 10^{-4}\,\text{m}^2,$

Plate separation: $d_{\text{initial}} = 3\,\mu\text{m} = 3 \times 10^{-6}\,\text{m}.$

For each modification, we compare the new factor:

Case C:

New dimensions:

$l = 6\,\text{cm} = 0.06\,\text{m}, \quad b = 5\,\text{cm} = 0.05\,\text{m}.$

New area:

$A_{\text{new}} = 0.06 \times 0.05 = 3 \times 10^{-3}\,\text{m}^2.$

Ratio of areas:

$\frac{A_{\text{new}}}{A_{\text{initial}}} = \frac{3 \times 10^{-3}}{3 \times 10^{-4}} = 10.$

The plate separation is unchanged:

$\frac{d_{\text{initial}}}{d_{\text{new}}} = \frac{3 \times 10^{-6}}{3 \times 10^{-6}} = 1.$

Overall factor:

$10 \times 1 = 10.$

Case E:

New dimensions:

$l = 5\,\text{cm} = 0.05\,\text{m}, \quad b = 2\,\text{cm} = 0.02\,\text{m}.$

New area:

$A_{\text{new}} = 0.05 \times 0.02 = 1 \times 10^{-3}\,\text{m}^2.$

Ratio of areas:

$\frac{A_{\text{new}}}{A_{\text{initial}}} = \frac{1 \times 10^{-3}}{3 \times 10^{-4}} \approx 3.33.$

New plate separation:

$d_{\text{new}} = 1\,\mu\text{m} = 1 \times 10^{-6}\,\text{m},$

so the ratio of separations is

$\frac{d_{\text{initial}}}{d_{\text{new}}} = \frac{3 \times 10^{-6}}{1 \times 10^{-6}} = 3.$

Overall factor:

$3.33 \times 3 \approx 10.$

Both Case C and Case E yield a capacitance that is 10 times the original value.

Thus, the correct modifications are those in Case C and Case E.