Energy Stored in a Capacitor 🔋

Move charge from one plate to the other and you do work. That work parks itself as electrostatic potential energy inside the capacitor.:contentReference[oaicite:0]{index=0}

1. Charging Up Step-by-Step 🏗️

  • Mid-way through charging, the positive plate holds charge \(Q’\) while the negative plate holds \(-Q’\). The potential difference sits at \(V’ = \dfrac{Q’}{C}\).:contentReference[oaicite:1]{index=1}
  • Shift a tiny bit of charge \(dQ’\) from the negative plate to the positive one; you must supply work
    \(dW = V’\, dQ’ = \dfrac{Q’}{C}\, dQ’\).:contentReference[oaicite:2]{index=2}
  • Add up all those little bits:
    \(W = \displaystyle\int_0^{Q} \dfrac{Q’}{C}\, dQ’ = \dfrac{Q^{2}}{2C}\).:contentReference[oaicite:3]{index=3}

2. Handy Energy Formulas 🧮

Rewrite the stored energy any way you like:

  • \(W = \dfrac{1}{2}CV^{2}\) — perfect when you know \(C\) and the voltage 🔌:contentReference[oaicite:4]{index=4}
  • \(W = \dfrac{Q^{2}}{2C}\) — handy if charge comes first ⚖️:contentReference[oaicite:5]{index=5}
  • \(W = \dfrac{1}{2}QV\) — a quick mix of charge and voltage 🔄:contentReference[oaicite:6]{index=6}

3. Energy Lives in the Field ⚡

Think beyond plates—the energy actually fills the electric field between them.

  • Surface charge density: \( \sigma = \dfrac{Q}{A}\).
  • Electric field between the plates: \(E = \dfrac{\sigma}{\varepsilon_{0}}\).:contentReference[oaicite:7]{index=7}
  • Stored energy for a parallel-plate capacitor (plate area \(A\), spacing \(d\)):
    \(U = \dfrac{1}{2}\varepsilon_{0}E^{2} A d\). 🍰 (Notice \(Ad\) is the field’s volume.):contentReference[oaicite:8]{index=8}
  • Energy density of any electric field:
    \(u = \dfrac{1}{2}\varepsilon_{0}E^{2}\). 📦:contentReference[oaicite:9]{index=9}

4. Quick Example 🚀

Single capacitor: A \(900\ \text{pF}\) capacitor hooked to a \(100\ \text{V}\) battery stores

\(Q = CV = 900\times10^{-12}\ \text{F}\times100\ \text{V} = 9\times10^{-8}\ \text{C}\).:contentReference[oaicite:10]{index=10}

Energy: \(W = \dfrac{1}{2}QV = 4.5\times10^{-6}\ \text{J}\). 🔋:contentReference[oaicite:11]{index=11}

Sharing charge: Disconnect the battery and connect this charged capacitor to an identical uncharged \(900\ \text{pF}\) capacitor. Each plate ends up with \(Q/2\); the common voltage drops to \(V/2\).

Total energy now: \(W_{\text{final}} = 2 \times \dfrac{1}{2}\left(\dfrac{Q}{2}\right)\left(\dfrac{V}{2}\right) = 2.25\times10^{-6}\ \text{J}\). 🍂 The “missing” half turns into heat and electromagnetic waves during the brief current rush.:contentReference[oaicite:12]{index=12}

Important Concepts for NEET 🎯

  1. The energy trio: \( \dfrac{1}{2}CV^{2} = \dfrac{Q^{2}}{2C} = \dfrac{1}{2}QV \)
  2. Energy density of an electric field: \(u = \dfrac{1}{2}\varepsilon_{0}E^{2}\)
  3. Connecting a charged capacitor to an identical uncharged one halves the stored energy—watch for “lost energy” questions
  4. For a parallel-plate capacitor, \(U = \dfrac{1}{2}\varepsilon_{0}E^{2}Ad\); area and spacing matter!

💪 Keep practicing and you’ll master capacitor energy in no time!