Energy Stored in a Capacitor
Capacitors are fundamental components in electrical circuits, capable of storing electrical energy in an electric field. Understanding how this energy is stored and calculated is crucial for mastering electromagnetism, especially for competitive exams like JEE.
The Work Done to Charge a Capacitor
Charging a capacitor involves moving charge from one plate to another against the electric field that builds up between the plates. This process requires work, and this work is stored as potential energy in the electric field.
Work done to charge a capacitor is stored as potential energy.
As charge accumulates on the capacitor plates, the voltage across them increases. To move additional charge, work must be done against this increasing voltage. This work is stored as potential energy.
Consider a capacitor with capacitance C. To charge it from zero charge to a final charge Q, we incrementally move small amounts of charge dq. The voltage across the capacitor at any instant is given by v = q/C, where q is the charge already accumulated. The work done to move the next increment of charge dq is dW = v dq = (q/C) dq. To find the total work done (W) to charge the capacitor to a final charge Q, we integrate this expression from q=0 to q=Q: W = ∫(q/C) dq from 0 to Q = (1/C) [q²/2] from 0 to Q = Q²/(2C). This work done is stored as potential energy (U) in the capacitor.
Work is done to move charge against the electric field that builds up between the capacitor plates as charge accumulates.
Formulas for Energy Stored
Using the relationship Q = CV, we can express the energy stored in a capacitor in several equivalent forms.
| Formula | Description |
|---|---|
| U = Q² / (2C) | Energy in terms of charge (Q) and capacitance (C). |
| U = ½ CV² | Energy in terms of capacitance (C) and voltage (V). This is derived by substituting Q = CV into U = Q²/(2C). |
| U = ½ QV | Energy in terms of charge (Q) and voltage (V). This is derived by substituting C = Q/V into U = ½ CV². |
The formula U = ½ CV² is often the most practical for calculations when the voltage across the capacitor is known.
Energy Density
Energy density refers to the amount of energy stored per unit volume of the electric field. For a parallel-plate capacitor, this can be expressed in terms of the electric field strength.
For a parallel-plate capacitor with plate area A and separation d, the capacitance is C = ε₀A/d. The electric field between the plates is E = V/d. The volume of the capacitor is V_vol = Ad. Substituting these into U = ½ CV²: U = ½ (ε₀A/d) (Ed)² = ½ ε₀ A E² d. The energy density (u) is U/V_vol = (½ ε₀ A E² d) / (Ad) = ½ ε₀ E². This formula shows that energy is stored uniformly throughout the volume occupied by the electric field.
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u = ½ ε₀ E²
Energy Stored in Dielectric-Filled Capacitors
When a dielectric material is inserted into a capacitor, its capacitance increases by a factor of the dielectric constant (κ). This affects the energy stored if the voltage is kept constant, or if the charge is kept constant.
If a capacitor is charged to a voltage V and then isolated (charge Q is constant), and a dielectric is inserted, the new capacitance is C' = κC. Since Q = C'V', and Q is constant, V' = Q/C' = Q/(κC) = V/κ. The new energy stored is U' = Q²/(2C') = Q²/(2κC) = U/κ. The energy decreases because the dielectric reduces the electric field, and the capacitor does work on the external circuit as it discharges slightly.
If a capacitor is charged to a voltage V and kept connected to the battery (voltage V is constant), and a dielectric is inserted, the new capacitance is C' = κC. The new energy stored is U' = ½ C'V² = ½ (κC)V² = κ (½ CV²) = κU. The energy increases because more charge is drawn from the battery to fill the larger capacitance at the same voltage.
Key Takeaways for JEE
Master the three formulas for energy stored: U = Q²/(2C), U = ½ CV², and U = ½ QV. Understand the concept of work done against the electric field. Be able to derive energy density. Crucially, differentiate between scenarios where charge is constant (isolated capacitor) versus voltage is constant (connected to battery) when a dielectric is introduced.
Learning Resources
Provides a clear explanation of the work done to charge a capacitor and the formulas for stored energy.
A comprehensive video tutorial covering capacitance, dielectrics, and energy storage with practical examples.
A YouTube video specifically tailored for JEE preparation, focusing on the energy stored in capacitors and related problems.
A detailed resource covering capacitance, energy storage, and related concepts with formulas and diagrams.
An article explaining capacitors and energy storage, often with solved examples relevant to competitive exams.
Explains the concept of energy stored in a capacitor, including derivations and formulas, suitable for exam preparation.
A step-by-step guide to understanding and calculating the energy stored in a capacitor.
Provides a broad overview of capacitors, including a section dedicated to energy storage and its mathematical formulation.
Official syllabus for JEE Main, which helps in understanding the scope and importance of topics like capacitance.
Offers solved problems related to energy stored in capacitors, which is essential for JEE practice.