Rucete ✏ AP Physics C In a Nutshell
12. Capacitors
This chapter introduces the definition of capacitors and capacitance, methods for calculating capacitance, capacitor combinations in series and parallel, energy stored in capacitors, and the effects of dielectrics.
Qualitative Introduction to Capacitors
• A capacitor consists of two conductive plates separated by an insulator or vacuum.
• Connecting plates to a battery causes one plate to become positively charged and the other negatively charged.
• Charge accumulation continues until the voltage across the capacitor equals the battery voltage, at which point current stops.
• Capacitors store electrical energy, which can be rapidly discharged to deliver high power.
Capacitance
• Capacitance (C) measures a capacitor’s ability to store charge: C = Q/V.
• Capacitance is a device property, independent of charge or voltage at a given time.
• Units: farads (F), though practical capacitors are usually in μF, nF, or pF ranges.
• In a charged capacitor, plates have equal but opposite charges; capacitance is always positive.
Calculating Capacitance Based on Geometry
• Assume a charge +Q on one plate and −Q on the other.
• Use Gauss’s law to find the electric field between plates.
• Integrate the electric field to find the voltage: |V| = Ed for parallel plates.
• Capacitance formula for a parallel plate capacitor: C = ε₀A/d.
Understanding the Dependence of Capacitance
• Increasing plate area A increases capacitance because more charge can be stored at the same voltage.
• Decreasing plate separation d increases capacitance because it reduces the voltage for a given charge.
Combinations of Capacitors in Series
• In series, capacitors share the same charge Q but different voltages across each.
• The total voltage is the sum of individual voltages: Vtotal = V₁ + V₂ + …
• The reciprocal of equivalent capacitance: 1/Ceq = 1/C₁ + 1/C₂ + …
• Series connection effectively increases the distance between plates, reducing overall capacitance.
Combinations of Capacitors in Parallel
• In parallel, capacitors share the same voltage but store different charges.
• The total charge is the sum of individual charges: Qtotal = Q₁ + Q₂ + …
• The equivalent capacitance is the sum: Ceq = C₁ + C₂ + …
• Parallel connection effectively increases the plate area, increasing overall capacitance.
Calculating Equivalent Capacitance of Complex Arrays
• Identify small groups of capacitors in series or parallel.
• Replace each group with its equivalent capacitance step-by-step.
• Continue simplifying until only one capacitor remains, representing the entire network.
Energy Stored in Capacitors
• Work is required to move charges onto capacitor plates because voltage increases as charge accumulates.
• Differential work: dW = V dQ = (Q/C)dQ.
• Total energy stored: U = ½QV = ½CV² = Q²/(2C).
• Energy can be expressed in terms of Q, V, or C depending on available data.
Capacitors with Dielectrics
• A dielectric material placed between plates increases capacitance by a factor of the dielectric constant κD (>1).
• Adding a dielectric reduces the electric field and voltage for the same amount of charge, thereby increasing capacitance.
• New capacitance: C = κD(ε₀A/d).
• Dielectrics induce opposite charges internally, partially canceling the field between plates.
Important Notes on Dielectrics
• Vacuum is considered a dielectric with κD = 1.
• Gauss’s law, as previously applied, is valid only in vacuum; modifications are necessary for materials with dielectrics.
• When calculating with a dielectric, first find vacuum capacitance and then multiply by κD.
Example: Capacitance of a Parallel Plate Capacitor with a Dielectric
• Starting from the vacuum capacitance C₀ = ε₀A/d.
• Inserting a dielectric modifies it to: C = κDε₀A/d.
• The dielectric reduces the effective electric field between plates, thus increasing the amount of charge that can be stored for a given voltage.
• Practical capacitors often use materials like plastic or ceramic as dielectrics to significantly boost capacitance without changing size.
Effect of Dielectrics in Disconnected Capacitors
• If a charged capacitor is disconnected from the battery before adding the dielectric, charge remains constant while voltage decreases.
• Since C = Q/V, an increase in capacitance results from the reduced voltage.
Effect of Dielectrics in Connected Capacitors
• If the capacitor remains connected to the battery while inserting a dielectric, the voltage stays the same, and the capacitor draws additional charge to maintain the new capacitance.
• Energy stored in the capacitor increases due to the additional charge stored.
In a Nutshell
Capacitors store electrical energy by accumulating opposite charges on two separated conductors. Their capacitance depends solely on their geometry and material properties. Capacitors combine differently in series and parallel, affecting total capacitance. Adding a dielectric increases capacitance by reducing the electric field and potential difference for a given amount of charge. Capacitors are essential components for energy storage, filtering, and timing circuits in practical electronic systems.