DC Electricity & Capacitors

DC Electricity & Capacitors NCEA AS3.6 Text Chapters 12-13

I1=2A I3=5A I2=3A Kirchhoff’s Laws • Current Law: The total current entering a junction in a circuit equals the total current leaving.

V2 V1 V3 Kirchhoff’s Laws • Voltage Law: The total of all the potential differences around a closed loop in a circuit is zero

-V +V Kirchhoff’s Laws • For voltage sources: • Going from + to – represents a loss of energy so potential difference is negative • Going from – to + represents a gain in energy so potential difference is positive

+V=+IR -V=-IR I I Kirchhoff’s Laws • For resistors: • Passing in the same direction as the current represents a loss of energy -V=-(IR) • Passing in the opposite direction to the current represents a gain in energy +V=+(IR)

a b 2A V? 6V 1Ω 3A 4Ω d c Kirchhoff’s Laws • To solve problems, follow a loop around a circuit… • a-b 0V • b-c +6V • c-d -8V (2Ax4Ω) • d-a +3V (3Ax1Ω) –V Then add up the p.d’s. 0V+6V-8V+3V-V=0 V=1V!

d a 5V I1? 2Ω 0.75A c b 4Ω I2? 8Ω V? e f Kirchhoff’s Laws • Harder…. • a-b 0V • b-c -0.75A x 4Ω = -3V • c-d -I1x 2 = -2I1V • d-a +5V • Adding: -3 -2I1 +5 = 0 • So I1 = 1A • Using the current law: • 1A = 0.75A + I2 • So I2 = 0.25A

Continued… b-c -0.75A x 4Ω = -3V c-f +0.25A x 8Ω = +2V f-e +V? e-b 0V Adding: -3V + 2V + V? = 0 So V = 1V d a 5V I1? 2Ω 0.75A c b 4Ω I2? 8Ω V? e f Kirchhoff’s Laws

Internal Resistance • All components have some resistance, including cells and meters • The potential difference (voltage) measured across a cell when no current is being drawn is called the E.M.F (electro-motive force) • When current flows, the voltage measured across the terminals of the cell will drop, because of the cell’s internal resistance.

V (V) EMF r EMF r A V I (A) Internal Resistance

Electrical Meters • Galvanometer – very sensitive meter that can be adapted to read either current or voltage • Must know 2 things about the galvanometer • The internal resistance • The current that causes full scale deflection of the needle If.s.d

Small current passes through G Rs r G Most current continues through circuit Electrical Meters • To make a voltmeter the galvanometer must be connected in series with a large resistor Rs

Rs G 200Ω 5mA 10V Electrical Meters • Example: Need to measure up to 10V with a meter that has 200Ω of internal resistance and a full scale deflection at 5mA. • V=IR • 10V= 0.005A x (200 + Rs) • Rs=1800Ω

Small current passes through G r G Rp Most current continues through circuit Electrical Meters • To make an ammeter the galvanometer must be connected in parallel with a very small resistor Rp

Electrical Meters 200Ω G 5mA Rp • Example: Need to measure up to 1A with a meter that has 200Ω of internal resistance and a full scale deflection at 5mA. • V (across meter) =IR = 0.005 x 200 = 1V • Current through Rp = 1A - 0.005A = 0.995A • Rp = V/I = 1V / 0.995A = 1Ω

Capacitors • A capacitor is a device that can store electric charge and release it later on. • They consist of two metal plates, separated by an insulating material called a dielectric. • Capacitors are used for • Tuning circuits eg radios • Flashing circuits eg camera flashes • Controlling alternating current

Capacitors • There are several types of capacitor:

Capacitors • Capacitors are charged by connecting them to a power supply or battery • This causes an accumulation of electrons on one plate (negative) and removal of an equal number of electrons from the other (positive)

Capacitors • When a capacitor is fully charged: • the current flow in the circuit stops • both plates have equal but opposite amount of charge on them • The voltage across the plates equals the supply voltage • An electric field exists between the plates. The strength of this field is given by: E=V/d

Capacitance C • The value of a capacitor is a measure of how much charge it can store per volt applied across the plates • The unit for capacitance is the farad F • 1F=1CV-1 • (Most capacitors are much less than 1F in size, it’s a big unit!)

Capacitance • The capacitance of a capacitor depends on 3 things: • The area of the plates which overlap (Cα A ie. The more area overlapping, the bigger the C) • The distance separating the plates (C α 1/d ie. The closer the plates the bigger the C) • What is used as the dielectric material.

Capacitance • These are summed up in the capacitor construction formula: A = Area of overlapping plates d = Distance separating plates εo = Absolute permittivity of free space (!!) – A constant = 8.84x10-12 Fm-1 εr =Dielectric constant (no units, it’s a factor)

Charging Capacitors • When the switch is closed, charge begins to build up on the plates. • The process continues until the cap has the same voltage across it as the power source

V Vmax t Charging capacitors • It’s easy at first because the plates are empty, but as they fill the stored charge starts to repel further charge and the rate decreases

I Imax t Charging Capacitors • As the rate of charge movement decreases, the current flow in the circuit decreases

V Vmax t Discharging Capacitors • The voltage will be at a max to begin with but will drop to zero as charge leaves the plates.

I Imax t Discharging Capacitors • The current will be large at first, but decrease as potential left to “push” decreases

Charge/Discharge Rates • The rate that a cap will charge or discharge depends on two things: • The size of the capacitor. The bigger the cap, the more charge it can store, so it will take longer to fill/empty. • The initial current in the circuit. This is determined by the resistance of the circuit. More R means smaller current, so longer to fill/empty.

Time Constant • This is defined as the time it takes the voltage/current to rise to 63% of it’s max value or for it to drop 63% from it’s max value • It is calculated using this formula: • A cap is considered fully charged or discharged after 3 time constants

Capacitor in Series • Joining capacitors in series:

-Q +Q V1 -Q V +Q V2 • Why? • The charge on each cap is the same. • The voltage across each cap adds to the supply voltage V=V1+V2

Capacitor Networks • Joining capacitors in parallel:

V Q2 Q1 • Why? • The voltage across each cap is the same as the supply. • The total charge stored is the sum of the charge in each cap. Q=Q1+ Q2 (NB. This is the opposite way round to resistors)

Energy in Capacitors • Capacitors store energy as electric potential energy. • The amount they store is half of the energy supplied by the battery. (The other half is dissipated as heat in the resistance of the circuit) • When a capacitor discharges, the energy is dissipated in the resistance of the circuit as heat, light etc.

For a cell: Each coulomb of charge gains V Joules of energy So the energy provided by the cell ΔE=VQ V ΔE= area =VQ Q Energy in Capacitors

V ΔE= area = ½ VQ Q Energy in Capacitors • For a capacitor: voltage increases gradually as cap charges until it equals the cell voltage • So the energy stored by the capacitor ΔE=1/2VQ

Energy in Capacitors • Some rearrangement and substitution puts this equation into a familiar form… • c.f. the formula for kinetic energy E=1/2mv2 • c.f. the formula for elastic potential energy E=1/2kx2

DC Electricity & Capacitors