Answer

Verified

388.4k+ views

**Hint:**The main use of the capacitor is to store electrical energy. A charged capacitor stores electrical energy in the electric field between its plates.

**Complete step by step answer:**

Consider capacitor of capacitance C being charged from a battery of emf V. Initially, its two plates are uncharged. The positive charge is transferred from the second plate to the first plate in a very small amount of $dq$ till the first plate acquires the final charge +Q and the second plate acquires the charge –Q.

At a certain instant t, during the process of charging, let q be the charge on the plate and V is the potential difference between the plates.

Then $q = CV$

$V = \dfrac{q}{C}$ ……………………… (1)

Where $C$ is the capacitance of the capacitor.

The amount of work done in transferring a small amount of charge $dq$ against the potential difference is,

$dw = Vdq$

Substitute equation (1) in above equation,

$dw = \dfrac{q}{C}dq$ ………………… (2)

The total work done in transferring charge Q is given by,

$W = \int\limits_0^W {dw} $

$W = \int\limits_0^W {\dfrac{q}{C}dq} $

$W = \dfrac{1}{C}\int\limits_0^W {qdq} $

Use this formula for integration,$\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} $

After integration we get,

$W = \dfrac{1}{C}\left[ {\dfrac{{{q^2}}}{2}} \right]_0^Q$

This is the amount of work done is stored as electrostatic potential energy U in the capacitor.

Thus energy stored in the capacitor is $U = \dfrac{1}{C}\left[ {\dfrac{{{Q^2}}}{2}} \right]$

$ = \dfrac{{{Q^2}}}{{2C}}$

We have $Q = CV$ substitute in above equation we get

$U = \dfrac{1}{2}C{V^2}$ These are the different kinds of equations of energy stored in a capacitor.

Energy density: energy stored per unit volume of the space between the plates is called energy density.

consider the separation $d$ of a parallel plate capacitor. Let us have $U = \dfrac{1}{2}C{V^2}$be the energy stored in the space of volume V=$A \times d$ between the plates.

Hence energy density, $u = \dfrac{{energy}}{{volume}}$

$ = \dfrac{{\dfrac{1}{2}C{V^2}}}{{Ad}}$……………………( 3)

But the capacitance $C = \dfrac{{{\varepsilon _0}A}}{d}$ and potential difference $V = Ed$

Substituting in equation (3)

$u = \dfrac{1}{2}\dfrac{{{\varepsilon _0}A}}{d} \times \dfrac{{{E^2}{d^2}}}{{Ad}}$

$u = \dfrac{1}{2}{\varepsilon _0}{E^2}$

This is a general expression which is applicable for electric fields due to any configuration of charges.

Energy stored before connection,

${U_1} = \dfrac{{{Q^2}}}{{2C}}$

${U_2} = 2\dfrac{{{{\left( {\dfrac{Q}{2}} \right)}^2}}}{{2C}}$

Energy stored after connection,

$ = \dfrac{{{Q^2}}}{{4C}}$

${U_2} < {U_1}$

From this we can say, storage of energy in the combination is less than that of the single capacitor.

**Note:**

Charging a capacitor means transferring electrons from one plate to another.

The capacitance of a capacitor depends on (i) the shape of its plates, (ii) separation between plates, and (iii) the nature of dielectric material between the plates.

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

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

Which are the Top 10 Largest Countries of the World?

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

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

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

The largest tea producing country in the world is A class 10 social science CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE