Answer

Verified

340.2k+ 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,

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

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

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

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

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE

State the laws of reflection of light

State and prove Bernoullis theorem class 11 physics CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE