A parallel plate capacitor having plate area 200 cm² and separation 2.0 mm holds a charge of 0.06 µC on applying a potential difference of 60 V. The dielectric constant of the material filled in between the plates is

The correct option is 11.3.

To find the dielectric constant of the material filled between the plates of the capacitor, we can use the relation between charge ($Q$), capacitance ($C$), and potential difference ($V$). The formula for the charge on a capacitor is given by:

$Q=CV$

For a parallel plate capacitor filled with a dielectric material of dielectric constant $K$, the capacitance $C$ is given by:

$C=\frac{K{\epsilon }_{0}A}{d}$

where:
• $K$ is the dielectric constant of the material,
• ${\epsilon }_0$ is the permittivity of free space (${\epsilon }_0\approx 8.85\times {10}^{-12}{\text{C}}^2{\text{N}}^{-1}{\text{m}}^{-2}$),
• $A$ is the area of each plate,
• $d$ is the separation between the plates.

Substituting the expression for $C$ into the charge equation, we get:

$Q=\frac{K{\epsilon }_{0}AV}{d}$

We can rearrange this formula to solve for the dielectric constant $K$:

$K=\frac{Q·d}{{\epsilon }_{0}AV}$

Let's convert the given values into standard SI units:
• Plate area, $A=200{\text{cm}}^2=200\times {10}^{-4}{\text{m}}^2=0.02{\text{m}}^2$,
• Separation, $d=2.0\text{mm}=2.0\times {10}^{-3}\text{m}$,
• Charge, $Q=0.06\text{μC}=0.06\times {10}^{-6}\text{C}=6.0\times {10}^{-8}\text{C}$,
• Potential difference, $V=60\text{V}$.

Now, substitute these values into the rearranged equation:

$K=\frac{(6.0\times {10}^{-8})\times (2.0\times {10}^{-3})}{(8.85\times {10}^{-12})\times \left(0.02\right)\times 60}$

Simplify the numerator:

$\text{Numerator}=1.2\times {10}^{-10}$

Simplify the denominator:

$\text{Denominator}=8.85\times {10}^{-12}\times 1.2=10.62\times {10}^{-12}=1.062\times {10}^{-11}$

Now divide the simplified numerator by the simplified denominator:

$K=\frac{1.2\times {10}^{-10}}{1.062\times {10}^{-11}}=\frac{1.2}{1.062}\times 10\approx 1.13\times 10=11.3$

Thus, the dielectric constant of the material filled in between the plates is approximately 11.3.