Potential Difference and Capacitance

The Potential difference (p.d.) between two points in an alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero field is the work done in bringing a charge of 1 alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero from one alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero to the other in an alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero field.



The unit of potential difference is the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero (symbol alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero) or from above J C ^-1.

The Volt

The potential difference between two points is one volt if one alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero of alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero is done when bringing a charge of alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero Coulomb from one point to another.

Note that potential difference is always between two points.

Relationship between Work, Charge and Voltage

Work = alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero × alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero:

W = alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero X alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero

Potential at a Point

The potential at a point refers to the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero done in bringing a positive charge from that alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero to alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.

The Earth is at alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero potential.

Capacitance

The symbol for capacitance is alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero

A capacitor is an electrical device used to store alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.

The Capacitance of a conductor is the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero of the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero on a conductor to its alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.

C = alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero / alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero

The unit of capacitance is the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero; symbol is alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.

The Parallel Plate Capacitor

Consider two oppositely charged parallel plates

The capacitance will increase if the common area between plates (A) alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero, or if the distance between plates (d) alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.

Therefore C ∝ A and C ∝ 1/ d C ∝ A / d C = ε ( A / d)

The proportional constant turns out to be ε (remember we came across this in the last chapter as part of Coulomb’s Law) and represents the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero of the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero between the two plates.


To Demonstrate the Factors affecting the Capacitance of a Parallel Plate Capacitance


1. Connect the two parallel plates to a digital multi-meter (DMM) set to read alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero. Note the capacitance.

2. Increase the distance between them – note that the capacitance alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.

3. Move one plate slightly to the side (decreasing the overlap area) – note that the alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero decreases.

4. Place different slabs of insulating material between the plates – note that the capacitance is alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero when nothing (air) is between the plates.


Energy stored in a Capacitor

W = ½ C alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero²

To show that a Charged Capacitor stores Energy (page 240)


1. Set up a capacitor with a bulb in alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.
2. Charge the capacitor by connecting the battery across it.
3. Remove the battery and connect the plates to a light bulb.
4. The bulb will flash as the capacitor discharges, showing that it stores energy.

Common Uses of Capacitors

1. The alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero on a camera
2. Smoothing out variations in direct current
3. Allows alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero current (a.c.) to pass through it but blocks alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero current (d.c.).
4. Used in circuits to only allow alternating current of a specific frequency to flow.



Capacitors – conduct a.c. but not d.c.

When a battery d.c and a capacitor are put in series there is a sudden movement of electrons, but because the circuit is not complete there is no further flow.

If the d.c. is now replaced with an a.c. supply then the plates are continuously charging and discharging; the time this takes (and therefore the frequency) depends upon the size of the plates .

Strictly speaking the current is not ‘passing through’ the capacitor, but merely acts as though it is.


When a charged capacitor is discharged through a lamp .

The electrons will flow and the bulb will alternatingCcapacitanceChargeCoulombdecreasesdirectearthelectricFFaradflashincreasesJoulelowestmediumonepermittivitypointpotentialQratioseriesVVoltVoltageworkzero.