Answer

Verified

417.9k+ views

**Hint**The capacitance of a capacitor is directly proportional to the dielectric constant and inversely proportional to the distance between the plates. Compare the two capacitances to each other by division.

Formula used: In this solution we will be using the following formulae;

\[C = \dfrac{{K\varepsilon A}}{d}\] where \[C\] is the capacitance of a capacitor, \[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.

**Complete Step-by-Step solution:**

According to the question, the first air is between the plates, hence, the dielectric constant is 1. Generally, the capacitance of a capacitor is given by

\[C = \dfrac{{K\varepsilon A}}{d}\] where\[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.

Hence, initially,

\[C = \dfrac{{\varepsilon A}}{d} = 8 \times {10^{ - 12}}\]

Then, capacitance after a dielectric material is added, and the distance between the plate is halved will be given as

\[{C_2} = \dfrac{{2K\varepsilon A}}{d}\]

Then, we compare the two capacitances by dividing, we have

\[\dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \div \dfrac{{\varepsilon A}}{d}\]

\[ \Rightarrow \dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \times \dfrac{d}{{\varepsilon A}}\]

Which by cancellation will give the expression

\[\dfrac{{{C_2}}}{C} = 2K\]

\[ \Rightarrow {C_2} = 2KC\]

By inserting all known values, we have

\[{C_2} = 2 \times 6 \times 8 \times {10^{ - 12}} = 9.6 \times {10^{ - 11}}F\]

**Note:**Noting that in the final expression the area, and permittivity of free space where absent, this implies that we can find the expression without their knowledge. Hence, alternatively, from the knowledge that the capacitance is proportional to dielectric constant but inversely to distance, we can just write that generally,

\[C = k\dfrac{K}{d}\] where \[k\] is an arbitrary constant. Hence,

\[\dfrac{{{C_2}}}{C} = k\dfrac{K}{{\dfrac{d}{2}}} \div k\dfrac{1}{d} = k\dfrac{{2K}}{d} \times \dfrac{d}{k}\]

\[ \Rightarrow {C_2} = 2KC\]

Watch videos on

A parallel plate capacitor with air between the plates has a capacitance of \[8\mu F\] (\[1\mu F \times {10^{ - 12}}\]). What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant 6?

Electrostatic Potential and Capacitance Class 12 Physics - NCERT EXERCISE 2.5 | Vishal Kumar Sir

Subscribe

Share

likes

118 Views

1 year ago

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which places in India experience sunrise first and class 9 social science CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE