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Energy Stored In A Capacitor | Energy Stored In A Capacitor Formula

Energy Stored In A Capacitor | Energy Stored In A Capacitor Formula :- Suppose a capacitor has capacitance $C$ and is connected to a battery, by which it is charged from zero potential to a potential . To determine the total energy stored in the capacitor, let us assume that initially both plates P₁ and P₂ are uncharged, and charge is transferred from plate P₂ to plate P₁ in very small amounts dq.

Since plate P₁ is at a higher potential and plate P₂ is at a lower potential, external work must be done to transfer each small amount of charge dq. This process continues until the charge on plate P₁ becomes +Q. According to the law of conservation of charge, at that time the charge on plate P₂ will be -Q.

Suppose that at some instant during charging, the charge on plate P₁ is +q and the charge on plate P₂ is -q. At this moment, the potential difference between the plates will be $\displaystyle \frac{q}{C}$ . The small amount of work done by the battery in transferring an infinitesimal charge dq is :-

$\displaystyle dW=\frac{q}{C}\times dq$

Total work done in charging a capacitor with charge +Q :-

$\displaystyle W=\int\limits_{0}^{Q}{dW=}\int\limits_{0}^{Q}{\frac{q}{C}\times dq=}\frac{1}{C}\int\limits_{0}^{Q}{qdq=}\frac{1}{C}\left[ \frac{{{q}^{2}}}{2} \right]_{0}^{Q}$

$\displaystyle \Rightarrow W=\frac{1}{C}\frac{{{Q}^{2}}}{2}$

Since the electrostatic force is conservative, the work done by the battery gets stored in the capacitor as its electrostatic potential energy (U).

$\displaystyle U=W=\frac{1}{C}\frac{{{Q}^{2}}}{2}$

Using Q = CV ,

$\displaystyle U=\frac{1}{C}\frac{{{\left( CV \right)}^{2}}}{2}=\frac{\text{1}}{2}C{{V}^{2}}$

Similarly using C = Q/V ,

$\displaystyle U=\frac{\text{1}}{2}\frac{Q}{V}{{V}^{2}}=\frac{\text{1}}{2}QV$

Hence, the energy stored in a capacitor,

$\displaystyle U=\frac{\text{1}}{2}\frac{{{Q}^{2}}}{C}=\frac{\text{1}}{2}C{{V}^{2}}=\frac{\text{1}}{2}QV$

Note :-

When a capacitor is connected to a battery, then :

Total energy supplied by the battery :

The battery supplies a total charge Q = CV to the capacitor, therefore the total energy supplied to the circuit by the battery (total work done by the battery) is :

W = Q⋅V = (CV)⋅V = CV²

Energy stored in capacitor :

$\displaystyle U=\frac{\text{1}}{2}C{{V}^{2}}$

Energy lost as heat :

Energy lost as heat (H) = Total energy supplied by the battery (W)− Energy stored in the capacitor (U)

$\displaystyle H=C{{V}^{2}}-\frac{\text{1}}{2}C{{V}^{2}}=\frac{\text{1}}{2}C{{V}^{2}}$

That is, half of the total energy is dissipated as heat.

✅ Why is heat produced?

When the capacitor is connected to the battery :

Initially, the capacitor has no charge → the electric current is high.
As the capacitor becomes charged → the potential increases → the electric current decreases.
This process slows down gradually, but current continues to flow throughout the charging process.
This current generates heat due to the resistance of the wires and other components.

➡ Conclusion: Half of the total energy supplied by the battery is stored in the capacitor, and the other half is dissipated as heat.

Energy Density of A Parallel Plate Capacitor (u)

(Energy Stored In A Capacitor | Energy Stored In A Capacitor Formula)

Energy Density : The energy stored per unit volume is called energy density.

If each plate of a parallel-plate capacitor has an area and the distance between the plates is , then the volume of the capacitor ( $V$ ) = Ad.

$\displaystyle Energy\text{ }Density(u)=\frac{Total\text{ }Energy\text{ }(U)}{Volume\text{ }(V)}=\frac{\frac{\text{1}}{2}C{{V}^{2}}}{Ad}$

$\displaystyle u=\frac{\text{1}}{2}\left( \frac{{{\varepsilon }_{0}}A}{d} \right)\left( \frac{{{V}^{2}}}{Ad} \right)$

If the value of the electric field between the plates of the capacitor is E , then

$\displaystyle V=Ed$

Therefore

$\displaystyle u=\frac{\text{1}}{2}\left( \frac{{{\varepsilon }_{0}}A}{d} \right)\left( \frac{{{E}^{2}}{{d}^{2}}}{Ad} \right)$

$\displaystyle u=\frac{\text{1}}{2}{{\varepsilon }_{0}}{{E}^{2}}$

Energy Stored In A Capacitor | Energy Stored In A Capacitor Formula

Energy Density of A Parallel Plate Capacitor (u)

(Energy Stored In A Capacitor | Energy Stored In A Capacitor Formula)

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