Capacitance — Revision Notes

⚡ 30-Second Revision

Capacitance Definition — $C = Q/V$ (Unit: Farad, F)
Parallel Plate Capacitor (Air/Vacuum) — $C = \frac{\epsilon_0 A}{d}$
Parallel Plate Capacitor (Dielectric) — $C = \frac{K\epsilon_0 A}{d} = \frac{\epsilon A}{d}$
Capacitors in Parallel — $C_{eq} = C_1 + C_2 + C_3 + \dots$
Capacitors in Series — $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$
Energy Stored — $U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$
Energy Density — $u = \frac{1}{2}\epsilon E^2$
Dielectric Constant (K) — $K = C_{dielectric}/C_{air}$
Effect of Dielectric (Battery Disconnected) — $Q$ constant, $V \downarrow$, $C \uparrow$, $E \downarrow$, $U \downarrow$
Effect of Dielectric (Battery Connected) — $V$ constant, $Q \uparrow$, $C \uparrow$, $E$ constant, $U \uparrow$

2-Minute Revision

Capacitance is the ability of a system to store electric charge, defined as $C=Q/V$. The SI unit is the Farad (F). A capacitor, typically two parallel plates separated by a dielectric, is designed for this.

For a parallel plate capacitor, $C = \frac{\epsilon_0 A}{d}$ in air. Inserting a dielectric material with dielectric constant $K$ increases capacitance to $C = \frac{K\epsilon_0 A}{d}$. Capacitors combine differently than resistors: in parallel, $C_{eq} = \sum C_i$; in series, $1/C_{eq} = \sum 1/C_i$.

A charged capacitor stores electrical potential energy in its electric field, given by $U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$. When analyzing dielectric insertion, remember two key scenarios: if the capacitor is disconnected from the battery, charge ($Q$) remains constant; if it remains connected, voltage ($V$) remains constant.

This distinction is crucial for determining changes in other quantities like $V, E,$ and $U$. Always convert units to SI before calculations.

5-Minute Revision

Capacitance is a measure of a capacitor's ability to store charge, expressed as $C = Q/V$. The Farad (F) is the unit, but microfarads ($mu\text{F}$), nanofarads ($ext{nF}$), and picofarads ($ext{pF}$) are more common.

For a parallel plate capacitor, its capacitance is geometrically determined by $C = \frac{\epsilon_0 A}{d}$ for vacuum/air. When a dielectric material with dielectric constant $K$ is inserted, the capacitance increases to $C' = K C_0 = \frac{K\epsilon_0 A}{d}$.

This is because the dielectric reduces the electric field, thus reducing the potential difference for the same charge.

Capacitors can be connected in two basic ways:

1
Parallel — $C_{eq} = C_1 + C_2 + C_3 + \dots$. Here, the voltage across each capacitor is the same, and the total charge is the sum of individual charges.
2
Series — $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$. Here, the charge on each capacitor is the same, and the total voltage is the sum of individual voltages.

The energy stored in a charged capacitor is $U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$. This energy resides in the electric field between the plates. The energy density is $u = \frac{1}{2}\epsilon E^2$.

Critical scenarios with dielectrics:

Battery disconnected (Q constant) — When a dielectric is inserted, $C$ increases, $V$ decreases, $E$ decreases, and $U$ decreases.
Battery connected (V constant) — When a dielectric is inserted, $C$ increases, $Q$ increases, $E$ remains constant, and $U$ increases.

Remember to convert all units to SI (e.g., $mu\text{F}$ to F, $ext{cm}^2$ to $ext{m}^2$, $ext{mm}$ to $ext{m}$) before performing calculations. Practice simplifying complex capacitor networks and applying energy conservation principles.

Prelims Revision Notes

1
Definition — Capacitance $C = Q/V$. SI unit: Farad (F). $1,\text{F} = 1,\text{C/V}$.
2
Parallel Plate Capacitor — $C = \frac{\epsilon_0 A}{d}$ (air/vacuum). $A$ is plate area, $d$ is separation.
3
Dielectric Effect — When dielectric (constant $K$) is inserted, $C' = KC = \frac{K\epsilon_0 A}{d}$. Dielectric strength is max E-field before breakdown.
4
Combinations

* Parallel: $C_{eq} = C_1 + C_2 + \dots$. Voltage is same across each. Total charge is sum. * Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$. Charge is same on each. Total voltage is sum.

1
Energy Stored — $U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$. Energy is in the electric field.
2
Energy Density — $u = \frac{1}{2}\epsilon E^2$.
3
Dielectric Scenarios

* Charged, then disconnected (Q constant): Insert dielectric $\implies C \uparrow, V \downarrow, E \downarrow, U \downarrow$. * Connected to battery (V constant): Insert dielectric $\implies C \uparrow, Q \uparrow, E$ constant, $U \uparrow$.

1
Common Mistakes

* Confusing series/parallel rules for capacitors vs. resistors. * Forgetting $1/2$ in energy formulas. * Incorrect unit conversions ($mu\text{F}$ to F, $ext{cm}^2$ to $ext{m}^2$, etc.). * Misinterpreting 'charge on capacitor' as net charge (it's magnitude on one plate).

1
Problem Solving Tips

* Simplify complex circuits step-by-step. * Identify symmetry or equipotential points. * Always use SI units for calculations. * For common potential problems, use conservation of charge: $V_{common} = \frac{C_1V_1 + C_2V_2}{C_1 + C_2}$.

Vyyuha Quick Recall

To remember capacitor combination rules (opposite of resistors): Capacitors Parallel Add, Capacitors Series Reciprocal. (CPA, CSR)