PHYC10004 Chap.4 Capacitors, Dielectrics and Stored Energy
Capacitors, Dielectrics and Stored Energy
A capacitor is two conductors holding equal and opposite charge, and its capacitance C = Q/V depends only on geometry. For parallel plates, Gauss's law gives a uniform field between them and C = ε₀A/d.
Weeks 3 and 4 of PHYC10004 develop this into the energy a capacitor stores, U = ½QV = ½CV² = Q²/2C, and the general result that any electric field carries an energy density ½ε₀E².
A dielectric between the plates polarises, reducing the field for a given free charge and raising the capacitance by its dielectric constant κ.
The central reasoning move of the chapter is to decide what is held fixed: a capacitor left connected to a battery keeps its voltage, while an isolated one keeps its charge, and the two cases respond in opposite ways to a dielectric or to moving the plates. The chapter also covers capacitors in series and in parallel, rules that are not printed on the formula sheet.
What this chapter covers
- 01
Capacitance and Geometry
- 02
Parallel Plate Capacitors
- 03
Energy Stored in a Capacitor
- 04
Electric Field Energy Density
- 05
Dielectrics and Polarisation
- 06
Connected Versus Isolated
- 07
Series and Parallel Capacitors
- 08
Dielectric Breakdown
Worked example · free
Share the voltage across two capacitors in series
- 1Series combination: 1/C = 1/2.0 + 1/3.0 = 5/6, so C = 1.2 μF.
- 1Charge: series capacitors carry the same charge, Q = CV = 1.2 × 10 = 12 μC on each.
- 1Voltages: V = Q/C gives 12/2.0 = 6.0 V across the 2.0 μF and 12/3.0 = 4.0 V across the 3.0 μF.
- 1Check: 6.0 V + 4.0 V = 10 V, the supply voltage.
- 1Energy: U = ½CV² = ½(1.2 × 10⁻⁶)(10)² = 6.0 × 10⁻⁵ J.
Key terms
- Capacitance
- Capacitance is the ratio of the charge on either conductor of a capacitor to the potential difference between them, C = Q/V, measured in farads.
- Dielectric Constant
- The dielectric constant κ is the factor by which a material filling a capacitor raises its capacitance, equal to the ratio of the material's permittivity to ε₀.
- Energy Density
- Energy density is the energy stored per unit volume of electric field, ½ε₀E² in vacuum and ½κε₀E² in a dielectric.
- Dielectric Strength
- Dielectric strength is the largest field a dielectric can withstand before it breaks down and conducts.
- Polarisation
- Polarisation is the slight separation of charge within the molecules of a dielectric in an applied field, which leaves bound charge on its faces that opposes the free charge.
- Equivalent Capacitance
- Equivalent capacitance is the single capacitance that stores the same charge at the same voltage as a network of capacitors.
Capacitors, Dielectrics and Stored Energy FAQ
What changes when a dielectric is inserted with the battery connected?
The battery holds the voltage fixed, so the capacitance, the charge and the stored energy all rise by the factor κ while the field between the plates stays V/d. Extra charge flows from the battery to make this happen.
What changes when a dielectric is inserted into an isolated capacitor?
The charge cannot change, so the voltage, the field and the stored energy all fall by the factor κ. The missing energy is the work done by the field in pulling the slab into the gap.
Are the series and parallel capacitor rules on the formula sheet?
No. The sheet gives C = Q/V, C = Aε₀/d, the energy forms and ε = κε₀, but you must recall that parallel capacitances add while series combinations add as reciprocals, the opposite pattern to resistors.
Why does a capacitor store energy at all?
Each extra charge is pushed onto a plate that already repels it, so charging takes work. That work is stored in the electric field between the plates and can be recovered, for example in a camera flash or defibrillator.
Exam move
Make a two-column table, battery connected and isolated, and fill it for three changes: inserting a dielectric, moving the plates closer and increasing the plate area. Then confirm each entry with numbers on one capacitor, computing the energy two ways each time. Practise networks by collapsing them step by step and labelling each capacitor with its charge and voltage.
Finally, rehearse the derivation of C = κε₀A/d from Gauss's law, with a diagram showing a uniform field between the plates and none outside, because past papers ask for exactly that chain.
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