Physics · Electricity and magnetism · Grade 8

Capacitor and capacitance

Why does a camera-flash capacitor take seconds to charge but discharge in an instant?

The situation

At a birthday party your friend is taking photos with a simple camera. The moment the button is pressed the flash lights up in an instant, but before the next flash you have to wait a few seconds until the ready light comes on. A single AA battery inside the camera cannot deliver such a powerful burst of light directly: the energy is first stored slowly in a capacitor. How does a capacitor store energy, and why does it release it all at once?

Topic

Capacitance and capacitors. Camera flash: the capacitor charges slowly and discharges in an instant

Key ideas

Capacitance C = ε₀εA/d
Energy W = CU²/2
RC charging and discharging

Formulas

C = \frac{q}{U}, C = \frac{ε_{0} ε S}{d}

The definition of capacitance and the capacitance of a parallel-plate capacitor; $ε_{0} = 8.85 \cdot 10^{- 12}$ F/m.

C_{b a l l} = 4 π ε_{0} ε R, C_{s p h} = \frac{4 π ε_{0} ε R_{1} R_{2}}{R_{2} - R_{1}}, C_{c y l} = \frac{2 π ε_{0} ε L}{\ln (R_{2} / R_{1})}

An isolated sphere, a spherical capacitor and a cylindrical (coaxial) capacitor.

W = \frac{C U^{2}}{2} = \frac{q^{2}}{2 C}, w = \frac{ε_{0} ε E^{2}}{2} = \frac{E D}{2}

The energy of a capacitor and the energy density of an electric field.

C_{p a r} = C_{1} + C_{2} + \dots, \frac{1}{C_{s e r}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \dots

Parallel (same voltage) and series (same charge) connection.

U (t) = U_{0} (1 - e^{- t / R C}), X_{C} = \frac{1}{ω C}

Charging in an RC circuit ( $τ = R C$ ) and capacitive reactance in an alternating-current circuit.

Theory

Conductor, dielectric and electrical capacitance

Conductors (metals, electrolytes) contain free charges: under the action of a field they shift and reduce the field inside the conductor to zero — the charge lies only on the surface, and the surface is an equipotential (the $E (r)$ graph in “Shape” mode is zero inside). In dielectrics (glass, plastic, air) there are almost no free charges. The electrical capacitance of an isolated conductor, $C = q / φ$ , is the charge needed to raise its potential by 1 V. Its unit is the farad: 1 F = 1 C/V; in practice mF ( $10^{- 3}$ ), µF ( $10^{- 6}$ ), nF ( $10^{- 9}$ ), pF ( $10^{- 12}$ ). The capacitance of a sphere is $C = 4 π ε_{0} ε R$ : a sphere of radius 10 cm in air has only 11 pF, while the whole Earth has about 0.71 mF. If the sphere is immersed in kerosene, the capacitance increases by a factor of ε = 2.1 — capacitance depends on the medium surrounding the conductor. For several conductors the charges depend linearly on the potentials: $q_{i} = \sum_{j} C_{i j} φ_{j}$ (the coefficients of capacitance).

Capacitors and their types

A capacitor is two conductors (plates) separated by a thin dielectric. The opposite charges on the plates cancel each other's field outside, so at the same voltage far more charge can be held than on a single conductor. Parallel-plate capacitor: $C = ε_{0} ε S / d$ ; with $S = 100$ cm², $d = 1$ mm and air, 88.5 pF. The formulas for spherical and cylindrical capacitors follow from Gauss's theorem; a coaxial cable is also a cylindrical capacitor (about 50–100 pF per metre). Types: paper, mica, ceramic, film, electrolytic (the dielectric is a very thin oxide layer: large capacitance, but polarised), the variable air-dielectric capacitor (tuning a radio receiver), and supercapacitors. Uses: the camera flash and the defibrillator (storing energy), smoothing voltage ripple in power supplies, RC timers, oscillating circuits, touchscreens.

A dielectric in a field: polarisation, the vectors P and D

In a field a dielectric becomes polarised: in atoms the electron cloud shifts (electronic polarisation), in ionic crystals the ions shift (ionic polarisation), and polar molecules (for example, water) turn along the field (orientation polarisation). As a result bound charges appear on the surfaces of the dielectric (the faint circles in the scene) and they weaken the field inside. The polarisation vector is $\vec{P} = ε_{0} χ \vec{E}$ ( $χ$ is the electric susceptibility), the relative permittivity is $ε = 1 + χ$ . The electric displacement vector $\vec{D} = ε_{0} \vec{E} + \vec{P} = ε_{0} ε \vec{E}$ starts only from free charges: $\oint \vec{D} \cdot d \vec{S} = q_{f r e e}$ , $div \vec{D} = ρ_{f r e e}$ — the Gauss theorem for dielectrics. In a partly inserted slab $E = U / d$ is the same in both parts, while $D$ and the free charge density are ε times greater in the dielectric — try switching between the “E” and “D” lines. The macroscopic $\vec{E}$ is the average of the very non-uniform microscopic field between molecules over a volume containing many molecules.

Boundary conditions, ferroelectrics, piezoelectrics

At the boundary between two dielectrics (with no free charge) the tangential component of $E$ and the normal component of $D$ are continuous: $E_{1 t} = E_{2 t}$ , $D_{1 n} = D_{2 n}$ , so the field lines are refracted: $\tan α_{1} / \tan α_{2} = ε_{1} / ε_{2}$ . Ferroelectrics (Rochelle salt, barium titanate BaTiO₃) have ε reaching thousands, the $P (E)$ relation shows hysteresis, and above the Curie point (about 120 °C for BaTiO₃) this property disappears; compact ceramic capacitors of large capacitance are made from them. Piezoelectrics (quartz, tourmaline, piezoceramics) become polarised when compressed: in a piezoelectric lighter a small hammer strikes the crystal and produces a voltage of several kilovolts and a spark; the reverse effect is used in the quartz watch resonator and in ultrasound sources. Electrets keep their polarisation for a long time (electret microphones). At high frequency the polarisation cannot keep up: the static ε of water is ≈ 81, while for light $n^{2} \approx 1.78$ ( $n = \sqrt{ε}$ — dispersion); in a medium that absorbs light the refractive index is a complex number.

The capacitor and field energy

When charging, each new charge $d q$ is moved against the voltage $U = q / C$ : $W = \int_{0}^{Q} \frac{q}{C} d q = \frac{Q^{2}}{2 C} = \frac{C U^{2}}{2}$ ; for a charged isolated body too, $W = q φ / 2$ . The energy is located in the field between the plates: per unit volume $w = ε_{0} ε E^{2} / 2 = E D / 2$ , and in a parallel-plate capacitor $W = w \cdot S d$ (the check row in the measurements). If the switch is open ( $Q$ = const), the work spent pulling the plates apart turns into field energy, while a dielectric slab is itself pulled inside and $W$ decreases (the energy graph). A 120 µF capacitor in a flash stores about 5.4 J at 300 V: charging takes seconds, discharge takes hundreds of microseconds — the flash power reaches tens of thousands of watts.

Connection, the RC circuit and alternating current

In a parallel connection the voltage across the capacitors is the same and the charges add: $C = C_{1} + C_{2} + \dots$ . In a series connection the charge is the same and the voltages add: $1 / C = 1 / C_{1} + 1 / C_{2} + \dots$ — the total capacitance is smaller than the smallest one, and the voltage divides in inverse proportion to capacitance. In an RC circuit a capacitor does not charge at once: as the current $I = (U_{0} - U) / R$ falls, charging slows down, $U (t) = U_{0} (1 - e^{- t / R C})$ ; at time $τ = R C$ it is charged to 63 %, at $5 τ$ to ≈ 99 %. In a flash the resistance of the charging circuit is kilohms, while that of the xenon lamp is ≈ 1 Ω — so the discharge is thousands of times faster. In alternating current, current flows through a capacitor: the capacitive reactance is $X_{C} = 1 / (ω C)$ , and the current leads the voltage by a quarter of a period (90°). With R and C in series, from the phasor diagram $Z = \sqrt{R^{2} + X_{C}^{2}}$ ; with a coil as well, Ohm's law gives $I = U / \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}$ , in complex form $Z = R + i (ω L - 1 / ω C)$ — see the “Alternating current” experiment for details.

Did you know?

The first capacitor — the “Leyden jar” — was invented independently by Ewald Georg von Kleist in 1745 and by Pieter van Musschenbroek in Leiden in 1746.
The capacitance of the whole Earth as an isolated conductor is about 0.7 mF — less than a common 1000 µF electrolytic capacitor used in electronics.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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