Capacitors

Capacitance

• Device designed to store charge

• When two metal plates are connected to a battery - electrons go from the negative terminal of the battery and build up on one of the plates. An equal number of electrons leave the other plate and return to the battery via the positive terminal

• Each plate has an equal and opposite charge

• The two parallel plates - conductors insulated from one another

Charging at constant current

• When plotting a graph of Charge against Potential Difference (of a capacitor) at a constant current, it is a straight line passing through the origin

• So the charge stored in the capacitor is proportional to the PD across the capacitor

Capacitance = Charge stored per unit of potential difference (measured in farads - F)

C = Q/V

Uses for capacitors

• Timing circuits (to automatically switch on and off)

• Pulse Producing circuits (to switch on and off repeatedly)

Energy Stored in a charged capacitor

• If capacitor charged - energy is stored as work done is needed to force the electrons onto/away from the plate

• Stored as Electric Potential Energy

E = ½ QV = ½ CV² = ½ Q²/C

Charging and Discharging a capacitor

Discharge through a fixed resistor

• Discharge current gradually decreases to 0

• Current decreases gradually because PD decreases as it loses charge. And since resistor is connected directly to capacitor, the resistor current decreases as PD decreases

• if plotting a graph of current or charge against time, they both follow an exponential decrease

• Therefore, in theory, the charge on the plates never become 0

For X = X₀ e^-r/RC

X can be either Q, I, or V

RC is the time constant for the circuit. at t = RC, the charge falls to 37% of the initial value - Resistance of Circuit x Capacitance (seconds - s)

Charging through a fixed resistor

• The charging current decreases as the capacitor charge and PD increase

• When the capacitor is fully charged, the PD across the capacitor is equal to the source PD, but the current is 0 because no more charge flows in the circuit

• If plotting a graph of Charge against Time, it is an exponential increase (with Q at 0.63 for t=RC). The curve flattens out at Q₀ = CV_{0 .} RC is therefore the time taken for the current to reach 63% of the final charge. THIS IS ALL THE SAME FOR CAPACITOR PD AGAINST TIME AS VC = Q (so are directly proportional and will follow same relationship)

• Current is given by the gradient of the Charge-Time graph

THE CHARGING EQUATION WORKS FOR V AND Q

Dielectrics

• Add a dielectric between the plates = increase the charge stored on the plates

• electrically insulating material

• Increase the ability of a parallel plate capacitor to store charge

• Each molecule of the dielectric becomes polarised

• so electrons are pulled slightly towards the positive plate

• the surface of the dielectric facing the the positive plate gains negative charge

• the other surface loses negative charge

More charge is stored because: the + side of the dielectric attracts more electrons from the battery onto the negative plate…the - side of the dielectric pushes electrons back to the battery from the positive plate.

REMEMBER:

• Charge builds up on the plates

• Electric field occurs between the plates

• Molecules are polarised so electrons align with positive end, and hence there is a positive and negative part of each molecule

• The molecules now have their own electric field, opposing the electric field that was produced by the charge on the capacitor plates

• Electric field strength therefore decreases

• AND electric field strength is relative to the V required to move charges between a capacitor (more E, then more V needed)

• So by decreasing E, you decrease the V needed to move a charge

• and via C = Q/V, if you decrease V, you increase C (increase capacitance/can store more charge per unit of pd)

Relative Permittivity

• The ratio of charge stored with the dielectric to the charge stored without the dielectric is the relative permittivity

ε_r = Q/Q_{0 ⁼} C/C₀

where Q = charge stored by parallel plate capacitor when the space is filled with the dielectric

Q₀ = Charge stored at the same PD when the space is empty

INCREASE SURFACE AREA OF EACH PLATE = INCREASE CAPACITANCE

DECREASE THE SPACING BETWEEN PLATES (d) = INCREASE CAPACITANCE

INCREASING THE RELATIVE PERMITTIVITY OF THE DIELECTRIC = INCREASE CAPACITANCE