Capacitor with Dielectric

Most capacitors have a dielectric (insulating solid or liquid material) in the space between the conductors. This has several advantages:

• Physical separation of the conductors. • Prevention of dielectric breakdown. • Enhancement of capacitance.

The dielectric is polarized by the electric ﬁeld between the capacitor plates.

tsl124 Parallel-Plate Capacitor with Dielectric (1)

The polarization produces a bound charge on the surface of the dielectric.

+q −q +q −q +q f f f b b −qf + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − + E − + − E + − + 0 − + − + − + − + − + − + − + − + − + − + − + − + − + − + −

~ ~ The bound surface charge has the effect of reducing the electric ﬁeld between the plates from E0 to E.

• A: area of plates • ±qb: bound charge on surface of dielectric ~ • d: separation between plates • E0: electric ﬁeld in vacuum • ±qf : free charge on plate • ~E: electric ﬁeld in dielectric tsl125 Parallel-Plate Capacitor with Dielectric (2)

~ ~ Use Gauss’ law to determine the electric ﬁelds E0 and E.

qf qf • Field in vacuum: E0A = ⇒ E0 = e0 e0A qf − qb qf − qb • Field in dielectric: EA = ⇒ E = < E0 e0 e0A V0 • Voltage: V = E d (vacuum), V = Ed = < V (dielectric) 0 0 κ 0

E0 qf Dielectric constant: κ ≡ = > 1. Permittivity of dielectric: e = κe0. E qf − qb tsl126 Dielectric Materials

• Dielectrics increase the capacitance: C/C0 = κ. • The capacitor is discharged spontaneously across the dielectric if the electric ﬁeld exceeds the value quoted as dielectric strength. tsl138 Impact of Dielectric (1)

What happens when a dielectric is placed into a capacitor with the charge on the capacitor kept constant?

vacuum dielectric

charge Q0 Q = Q0 E0 electric ﬁeld E E = < E 0 κ 0 V0 voltage V V = < V 0 κ 0 Q0 Q capacitance C0 = C = = κC0 > C0 V0 V 2 2 Q0 Q U0 potential energy U0 = U = = < U0 2C0 2C κ (0) (0) 1 1 u (0) energy density u = e E2 u = eE2 = E < u E 2 0 0 E 2 κ E

tsl127 Impact of Dielectric (2)

What happens when a dielectric is placed into a capacitor with the voltage across the capacitor kept constant?

vacuum dielectric

voltage V0 V = V0

electric ﬁeld E0 E = E0 Q0 Q capacitance C0 = C = = κC0 > C0 V0 V charge Q0 Q = κQ0 > Q0 1 1 potential energy U = C V2 U = CV2 = κU > U 0 2 0 0 2 0 0 (0) 1 1 (0) (0) energy density u = e E2 u = eE2 = κu > u E 2 0 0 E 2 E E

tsl128 Stacked Dielectrics

Consider a parallel-plate capacitor with area A of each plate and spacing d.

e0A • Capacitance without dielectric: C = . 0 d

• Dielectrics stacked in parallel: C = C1 + C2 A/2 A/2 with C = κ e , C = κ e . 1 1 0 d 2 2 0 d 1 ⇒ C = (κ + κ )C . 2 1 2 0

1 1 1 • Dielectrics stacked in series: = + C C1 C2 A A with C = κ e , C = κ e 1 1 0 d/2 2 2 0 d/2 2κ1κ2 ⇒ C = C0. κ1 + κ2

tsl129 Lateral Force on Dielectric

Consider two charged capacitors with dielectrics only halfway between the plates.

In conﬁguration (a) any lateral motion of the dielectric takes place at constant voltage across the plates.

In conﬁguration (b) any lateral motion of the dielectric takes place at constant charge on the plates.

Determine in each case the direction (left/zero/right) of the lateral force experienced by the dielectric.

(a) (b)

tsl130 Geiger Counter

Radioactive atomic nuclei produce high-energy particles of three different kinds:

α β γ

• α-particles are 4He nuclei. amplifier β • -particles are electrons or loudspeaker positrons. • γ-particles are high-energy photons. low−pressure inert gas

• Free electrons produced by ionizing radiation are strongly accelerated toward the central wire. • Collisions with gas atoms produce further free electrons, which are accelerated in the same direction. • An avalanche of electrons reaching the wire produces a current pulse in the circuit.

tsl123 Capacitor Circuit (4)

Connect the three capacitors in such a way that the equivalent capacitance is Ceq = 2µF. Draw the circuit diagram.

1µF 3µF 5µF

2µF

tsl117