The capacitance of a parallel plate capacitor with air as medium is $6\, \mu F$. With the introduction of a dielectric medium, the capacitance becomes 30 $\mu F.$ The permittivity of the medium is :
$(\epsilon_0 = 8.85 \times 10^{-12} C^2N^{-1}m^{-2}$)
- $0.44 \times 10^{-13} C^2 N^{-1}m^{-2}$
- $1.77 \times 10^{-12} C^2 N^{-1}m^{-2}$
- $0.44 \times 10^{-10} C^2 N^{-1}m^{-2}$
- $5.00 \,C^2 N^{-1}m^{-2}$
The Correct Option is A
Solution and Explanation
To find the permittivity of the dielectric medium used in the parallel plate capacitor, we need to understand the relationship between the initial and final capacitance, and the properties of the dielectric material introduced.
The capacitance of a parallel plate capacitor with air (or vacuum) as the dielectric medium is given by:
\(C_0 = \frac{\epsilon_0 \cdot A}{d}\)
where \(\epsilon_0 = 8.85 \times 10^{-12} \, C^2 N^{-1}m^{-2}\) is the permittivity of free space, \(A\) is the plate area, and \(d\) is the separation between the plates.
When a dielectric medium is introduced, the capacitance increases by a factor known as the dielectric constant (or relative permittivity \(\kappa\) of the medium:
\(C = \kappa \cdot C_0\)
We are given:
- \(C_0 = 6 \, \mu F = 6 \times 10^{-6} \, F\)
- \(C = 30 \, \mu F = 30 \times 10^{-6} \, F\)
Thus, the dielectric constant \(\kappa\) can be calculated as:
\(\kappa = \frac{C}{C_0} = \frac{30 \times 10^{-6}}{6 \times 10^{-6}} = 5\)
The permittivity of the dielectric medium, \(\epsilon\), is given by:
\(\epsilon = \kappa \cdot \epsilon_0\)
Substituting the known values:
\(\epsilon = 5 \cdot 8.85 \times 10^{-12} \, C^2 N^{-1}m^{-2} = 44.25 \times 10^{-12} \, C^2 N^{-1}m^{-2}\)
Converting this into the required form:
\(\epsilon = 4.425 \times 10^{-11} \, C^2 N^{-1}m^{-2} = 0.4425 \times 10^{-10} \, C^2 N^{-1}m^{-2}\)
Upon converting this to the closest given option:
- \(0.44 \times 10^{-10} \, C^2 N^{-1}m^{-2}\)
Thus, the permittivity of the medium is 0.44 × 10-10 C2 N-1m-2, which matches with \(0.44 \times 10^{-10} \, C^2 N^{-1}m^{-2}\) from the given options.
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