CAPACITANCE

• An electrical component that stores charge

.

• Separates charge into -Q and +Q

.

• Two metal plates separated by a dielectric

• An insulator

.

• Air/ paper/ ceramic etc.

Capacitor connected to a power supply

Electrons flow from the power supply for a very short time

Cannot travel between the plates because of the dielectric

Deposited onto first plate and removed from second plate simultaneously

First plate gains electrons/ aquires a net negative charge

Second plate loses in electrons/ aquires a net positive charge

Current same at all points and charge conserved

First plate stores charge -Q and second plate stores charge +Q

The charge stored per unit potential difference across a capacitor

Q (charge) (coulombs) = V (potential difference) (volts) x C (capacitance) (farads)

Total C > individual Cs

∴ more Q stored for a given V

Total C < individual Cs

∴ less Q stored for a given V

The total charge stored is equal to the sum of the individual charges

Q = Q₁ + Q₂ + ...

.

The potential difference across each capacitor is the same

Q = VC

VC = VC₁ + VC₂ + ...

.

The potential difference cancels out

C = C₁ + C₂ + ...

Kirchhoff's second law: sum of Es = sum of Vs in a closed loop

V = V₁ + V₂ + ...

.

The charge stored by each capacitor is the same

Q = Q₁ + Q₂ + ...

Q/C = Q/C₁ + Q/C₂ + ...

.

The charge cancels out

1/C = 1/C₁ + 1/C₂ + ...

An electron moves towards the negative plate of the capacitor being charged

Experiences a repulsive electrostatic force from all the electrons already on the plate

External work must be done to

• push an electron onto the negative plate
• pull an electron off the positive plate

Provided by the power supply connected to the capacitor

Energy stored in the capacitor comes from the energy of the power supply

• Straight line

• Constant, positive gradient

• Through the origin

.

ΔW (small amount of work done)

......= V (insignificant change in potential difference)

............x ΔQ (small increase in charge stored in capacitor)

.

ΔW = area of rectangular *****

W (total work done) = sum of areas of rectangular *****s

Work done on charges = energy stored in capacitor

.

W = area under graph

= ½QV

.

W = ½QV

= ½(VC) x V

= ½V²C

.

W = ½QV

= ½Q x (Q / C)

= ½Q² / C

A constant-ratio process in which a quantity decreases by the same factor in equal time intervals

Switch is closed

Capacitor is fully charged

V across capacitor and resistor = V_o (V across power supply)

.

Switch is opened

V across capacitor or resistor = V_o

Q stored in capacitor = V_oC

I in resistor = V_o / R

.

Capacitor discharges through resistor

V across/ Q stored in capacitor decreases exponentially with time

V, Q and I = zero

V, Q and I decrease exponentially with t after the switch is opened

.

V = V_oe^{- t / CR}

Q = Q_oe^{- t / CR}

I = I_oe^{- t / CR}

.

V / Q / I = potential difference / charge / current at any time (t)

V_o / Q_o / I_o = maximum potential difference / charge / current at t = 0

e = exponential function (2.718...)

• τ (time constant) (seconds) = C (capacitance) (farads) x R (resistance) (ohms)

.

• t for the V/ Q / I to decrease to e^-1 of its initial value after the switch is opened

I = V / R

= Q / CR (as V = Q / C)

.

I = - ΔQ / Δt ( - shows that Q decreases exponentially with t)

.

Q / CR = - ΔQ / Δt

Switch is open

Capacitor is uncharged

Power supply provides a constant V_o

.

Switch is closed

Maximum current in the circuit

Capacitor charges

.

Kirchhoff's second law: V_R + V_C = V_o

V_R decreases/ V_C increases exponentially with time

.

Capacitor is fully charged

V across capacitor = V_o

V_R and I = zero

I = I_oe^{- t / CR}

∴ V_R = V_oe^{- t / CR}

.

At t, V_R + V_C = V_o

∴ V_C = V_o - V_oe^{- t / CR}

= V_o (1 - e^{- t / CR})

• For the resistor, V = IR

.

• For the capacitor, Q = CV

.

• For the capacitor, x = x_o (1 - e^{- t / CR}), where x = V or Q

.

• In the circuit, I = I_oe^{- t / CR}

.

• At t, V_R + V_C = V_o

Power = energy stored / time

P = (1/2 x V² x C) / t

Domestic electricity supplied as alternating current

Direction of current changes as supply voltage cycles from positive to negative and back

Diode allows current in one direction only

Capacitor smooths out output voltage and prevents output voltage from consisting of positive cycles only

Output voltage becomes almost direct and constant

Can become more constant by making the time constant much greater than the time period