19.1 Capacitors and capacitance

Learning objectives

19.1.1Capacitance, as applied to both isolated spherical conductors and to• define capacitance, as applied to both isolated spherical conductors and to parallel plate capacitors 19.1.2C = Q / V• recall and use C = Q / V 19.1.3C = Q / V, formulas for the combined capacitance of capacitors in series and• derive, using C = Q / V, formulas for the combined capacitance of capacitors in series and in parallel 19.1.4The capacitance formulas for capacitors in series and in parallel• use the capacitance formulas for capacitors in series and in parallel

$capacitance C=Q/V, measured in farads (F), where 1 F=1 C V⁻¹$

Arrangement	Q means	V means
isolated conducting sphere	charge on the sphere	potential of its surface relative to infinity
parallel-plate capacitor	magnitude of charge on either one plate (+Q or -Q)	potential difference between the plates

Applying a p.d. to separated conducting plates moves charge from one plate to the other, creating equal and opposite plate charges and storing energy in the electric field.

Capacitance depends on conductor geometry and the material between conductors. A larger isolated sphere has greater C; parallel-plate C changes with plate area, separation and dielectric.

Capacitance is not a fixed number of coulombs that a device can hold. For unchanged geometry and dielectric, increasing V increases Q proportionally but does not itself increase C.

For a fixed capacitor, Q=CV. The slope of a Q–V graph is C.

Use farads, coulombs and volts; identify whether voltage is across the capacitor and account for dielectric or geometry changes before treating C as constant.

A 220 μF capacitor charged to 5.0 V stores 1.1×10⁻³ C.

The capacitor does not store a fixed amount of charge independent of voltage; C is the proportionality constant.

For parallel capacitors, every capacitor has the same p.d. V, while charge supplied by the source divides: Q_T=Q_1+Q_2+... .

$C_TV=C_1V+C_2V+... Therefore C_T=C_1+C_2+...$

For initially uncharged capacitors in series, each capacitor acquires the same charge magnitude Q, while p.d.s add: V_T=V_1+V_2+... .

$Q/C_T=Q/C_1+Q/C_2+... Therefore 1/C_T=1/C_1+1/C_2+...$

$For two in series: C_T=C_1C_2/(C_1+C_2)$

Connection	Shared/conserved fact	Bound check
parallel	same V; charges add	C_T is greater than the largest C
series	same charge magnitude; p.d.s add	C_T is less than the smallest C

The formulas are consequences, not resistor rules to swap by memory. A derivation must state the common charge or voltage and the corresponding additive total before substituting Q=CV.

Connection	Combined capacitance	Charge relation	p.d. relation
parallel	C_T=ΣC	Q_T=ΣQ_i	V_T=V_1=V_2=...
series	1/C_T=Σ(1/C)	Q_T=Q_1=Q_2=...	V_T=ΣV_i

Step	Action
1	identify one innermost series or parallel group
2	replace it by its equivalent capacitance
3	repeat to find C_T
4	use Q_T=C_TV_T, then expand backward using shared Q or V

A 20 μF and 10 μF pair in parallel gives 30 μF. In series with 45 μF, C_T=(1/30+1/45)⁻¹=18 μF.

Across a 12 V supply, that network stores series charge Q_T=(18 μF)(12 V)=216 μC. The 30 μF branch group has V=216 μC/30 μF=7.2 V, so its 20 μF and 10 μF capacitors carry 144 μC and 72 μC respectively.

Check topology at each step: equal charge applies only to elements in the same series chain, and equal p.d. only to branches across the same two nodes. Also check that series C_T is below the smallest member and parallel C_T above the largest.

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—19.2 Energy stored in a capacitorRead

19.1 Capacitors and capacitance

Learning objectives

Capacitance is charge per potential for a conductor or capacitor

Capacitance is C=Q/V, so stored charge rises linearly with voltage for a fixed capacitor

Derive capacitor combination formulas from charge and voltage constraints

Reduce capacitor networks, then recover each charge and p.d.

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