A parallel plate capacitor was made with two rectangular plates, each with a length of \(l=3 \mathrm{~cm}\) and breath of \(\mathrm{b}=1 \mathrm{~cm}\). The distance between the plates is \(3 \mu \mathrm{~m}\). Out of the following, which are the ways to increase the capacitance by a factor of 10 ?
A. \(l=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
B. \(l=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}\)
C. \(l=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}\)
D. \(l=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m}\)
E. \(l=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
Choose the correct answer from the options given below:

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Answer & Solution

Correct Answer

(D) \(C\) and \(E\) only

Step-by-step Solution

Detailed explanation

We know \(C=\frac{A \varepsilon_0}{d}\)
\(=\frac{b \ell \varepsilon_0}{d}\)
So to increase the capacitance by 10 factor \(\left(\frac{A}{d}\right)\) has to increase by 10 fator.
For option (A) \(\quad C^{\prime}=\frac{(30 \ell) b \varepsilon_0}{\left(\frac{d}{3}\right)}=30 \mathrm{C}\)
For option (B) \(\quad C^{\prime}=\frac{\ell b \varepsilon_0}{10 d}=\frac{C}{10}\)
For option (C) \(\quad C^{\prime}=\frac{(2 \ell) 5 b \varepsilon_0}{d}=10 \mathrm{C}\)
For option (D) \(\quad C^{\prime}=\frac{\left(\frac{\ell}{3}\right) b \varepsilon_0}{\left(\frac{10 d}{3}\right)}=\frac{C}{10}\)
For option (E) \(\quad C^{\prime}=\frac{\left(\frac{\ell}{3}\right) 5(2 b) \varepsilon_0}{\left(\frac{d}{3}\right)}=10 C\)
Clearly (C) and (E) are the situation for 10 C

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