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

Kirchhoff’s Laws Current Law: The total current entering a junction in a circuit equals the total current leaving. I 1 =2A I 2 =3A I 3 =5A

Kirchhoff’s Laws Voltage Law: The total of all the potential differences around a closed loop in a circuit is zero V1V1 V3V3 V2V2

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

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

Kirchhoff’s Laws To solve problems, follow a loop around a circuit… a-b0V a-b0V b-c+6V b-c+6V c-d-8V (2Ax4Ω) c-d-8V (2Ax4Ω) d-a+3V (3Ax1Ω) –V d-a+3V (3Ax1Ω) –V Then add up the p.d’s. 0V+6V-8V+3V-V=0V=1V! V? 6V 4Ω4Ω 1Ω1Ω ab cd 2A 3A

Kirchhoff’s Laws Harder…. a-b0V a-b0V b-c-0.75A x 4Ω = -3V b-c-0.75A x 4Ω = -3V c-d-I 1 x 2 = -2I 1 V c-d-I 1 x 2 = -2I 1 V d-a+5V d-a+5V Adding: -3 -2I 1 +5 = 0 Adding: -3 -2I 1 +5 = 0 So I 1 = 1A So I 1 = 1A Using the current law: 1A = 0.75A + I 2 1A = 0.75A + I 2 So I 2 = 0.25A So I 2 = 0.25A 4Ω4Ω 2Ω2Ω 8Ω8Ω 5V V? 0.75A I1?I1? I2?I2? a b c d e f

Kirchhoff’s Laws Continued… b-c-0.75A x 4Ω = -3V b-c-0.75A x 4Ω = -3V c-f+0.25A x 8Ω = +2V c-f+0.25A x 8Ω = +2V f-e+V? f-e+V? e-b0V e-b0V Adding: -3V + 2V + V? = 0 Adding: -3V + 2V + V? = 0 So V = 1V So V = 1V 4Ω4Ω 2Ω2Ω 8Ω8Ω 5V V? 0.75A I1?I1? I2?I2? a b c d e f

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.

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

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 internal resistance The current that causes full scale deflection of the needle I f.s.d The current that causes full scale deflection of the needle I f.s.d

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

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 V=IR 10V= 0.005A x (200 + R s ) 10V= 0.005A x (200 + R s ) R s =1800Ω R s =1800Ω G RsRs 200Ω 10V 5mA

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

Electrical Meters 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 = x 200 = 1V V (across meter) =IR = x 200 = 1V Current through R p = 1A A = 0.995A Current through R p = 1A A = 0.995A R p = V/I = 1V / 0.995A = 1Ω R p = V/I = 1V / 0.995A = 1Ω G RpRp 200Ω 5mA

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 Tuning circuits eg radios Flashing circuits eg camera flashes Flashing circuits eg camera flashes Controlling alternating current Controlling alternating current

Capacitors There are several types of capacitor: Non-polar Electrolytic Variable

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 the current flow in the circuit stops both plates have equal but opposite amount of charge on them both plates have equal but opposite amount of charge on them The voltage across the plates equals the supply voltage 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 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 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) 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. 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 ε o = Absolute permittivity of free space (!!) – A constant = 8.84x Fm -1 ε r =Dielectric constant (no units, it’s a factor) Air1 Oiled paper 2 Polystyrene2.5 Glass6 Water80

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

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 V t V max

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

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

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

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 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. 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:

Why? The charge on each cap is the same. The voltage across each cap adds to the supply voltage V=V 1 +V 2 V V2V2 V1V1 +Q -Q

Capacitor Networks Joining capacitors in parallel:

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=Q 1 + Q 2 V Q2Q2 Q1Q1 (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.

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

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 V Q ΔE= area = ½ VQ

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