1. o Aim of the lecture  Basic Current in terms of charge/time  Circuit Elements - Outline  Capacitance Energy storage Dielectrics Geometries o Main learning outcomes  familiarity with  Current and Voltage  Capacitance  Calculation of capacitance Lecture 4

2. Reminder: o Objects have a property called ‘charge’  Electrons have one unit of charge = 1.6 x 10 -19 C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o Moving charge is a current  If one coulomb per second passes through a loop Then the current through the loop is 1 Amp  It takes 6.3 x 10 18 electrons/second to make a 1A current

3. o There are three basic quantities in electrical circuits o Charge not really used very much o Voltage (which is the electrostatic potential) o Current (which is the movement of charge) o We will make an analogy to clarify o Consider a water barrel, pipe and water Reminder:

4. The water is like charge The volume of water per second is like current The height of the barrel is like Voltage If the pipe is unchanged, then more height = more water/sec more voltage = more current A pipe is like an electrical wire

5. To make this analogy work really well need to add a sump & pump: Charge is like water in the barrel water flows out of the barrel at a rate determined by the height of the barrel Voltage is like height Current is like flow rate of water Pump returns water to barrel Water driven tool

6. Pump returns water to barrel Note that the amount of water in the Barrel does not change. The pump will not let the water collect in the sump water can only flow round the circuit The amount of charge cannot change It can only move round the circuit Charge cannot collect anywhere

7. Pump returns water to barrel This is the water equivalent of a voltage source. For water, the potential is the height For charge, the potential is the Voltage H=h 0 H=0  H=h 0 V=V 0 V=0  V=V 0

8. Pump returns water to barrel A voltage source is drawn like this: This symbol represents this (or the charge equivalent of it)

9. Electrons are driven from the shorter bar by the voltage generated inside the voltage source. In a battery the voltage is generated chemically o Charge cannot flow, ie there is no current  unless there is a complete circuit  charge cannot collect anywhere  a loop is needed. Note: o The wire is already full of electrons Like a pipe already full of water o when the current starts to flow, o all the electrons in the wire move at the same time, The source of electrons is called the ‘negative terminal’ Or it can be called the ‘cathode’ - +

10. DO NOT BUILD A CIRCUIT LIKE THE ONE JUST SHOWN! (It would not be good for the voltage source)

11. The current is electrons moving round the circuit NOTE that this is called the electron current - +

12. The current is electrons moving round the circuit BUT ‘conventional current’ is considered to flow from the positive to the negative terminal. This is the current. This is definitely confusing, but is the result of history Current was discovered before electrons and the physicists thought that the charge carrier was positive. So we are stuck with a ‘wrong’ convention, where current flows from the +ve terminal to the –ve. - +

13. oWe considered the electric potential  in free space a voltage source produces potential lines  the potential energy depends on location  as a charge moves through a field from V1 to V2  it gains energy in the form of kinetic energy

14. + Voltage = V 0 Voltage = 0 + D h So the potential Energy,U U = q {V 0 /D} h compare with gravity U = m {Gm e /r e } h

15. o We considered the electric potential  in free space a voltage source produces potential lines  the potential energy depends on location  as a charge moves through a field from V1 to V2  it gains energy in the form of kinetic energy o However if a voltage is applied across a material then accelerated electrons collide with atoms  Loose energy  Accelerate again  Loose energy  Accelerate again  ….repeat…  They have a constant AVERAGE speed  The energy they gain is lost to collisions with atoms  This ends up as heat in the material

16. oDifferent conductors have different collision properties oThe result is that there is a fixed ratio between current and voltage oThis relationship  is linear  is called Ohm’s Law  is written V = I R V is the voltage across the material I is the current that flows and R is a constant, called resistance, which characterises  the properties of the material  the size and shape of the material  it is just a single number  the unit is the Ohm,  V = I R We will come back to Ohm’s law for a more detailed study

17. Capacitance - + If a voltage source is connected across a pair of plates, then charge will flow for a while and then stop It cannot keep flowing as there is no circuit Schematic representation

18. Electrons leave one plate The same number arrive on the other Capacitors store energy in the electric field between the plates NOTE: most books will talk about capacitors storing ‘charge’ there is actually no NET charge stored, so this is wrong What is true is that energy is stored because charges move from one place to another inside a capacitor, using an external circuit

19. Capacitance - + Schematic representation oThe total charge, Q, which flows from one plate to the other depends on: the voltage, V, applied size of plates separation between plates the material between the plates V

20. Capacitance - + Schematic representation V All the properties of the plate are summarised in one quantity, called the CAPACITANCE, C And Q = C V The unit of C is the Farad, F

21. Capacitance Q = C V 1 farad = 1F = 1 Coulomb per Volt = 1 C/V The farad is a very large unit, normally we use  F, nF and pF

22. Capacitance Q = C V o Calculating the value of the Capacitance from  Geometry  Material q =  0 E.dA ∫ Where q is the charge inside the surface E is the electric field A is an area element Recall Gauss’ Law:

23. Consider the simple parallel plate geometry: o The metal plates are equipotentials themselves A voltage difference applied between two plates Produces an electric field pointing from one plate to the other A constant field gradient between the plates Voltage=0 Voltage=V 0 E =  V/d d = V 0 /d We assume that the plates are large Compared with d Area = A Separation = d << {A} 1/2

24. Voltage=0 Voltage=V 0 E =  V/d d = V 0 /d We assume that the plates are large Compared with d Area = A Separation = d << {A} 1/2 So the integral becomes: Q =  0 V 0 dA ∫ d q =  0 E.dA ∫ Consider a Gaussian surface enclosing one of the plates, so just the charge on one plate Which finally gives us Q =  0 V 0 A/d so C =  0 A/d

25. However if we fill the gap between plates with an insulator A new constant must be introduced to represent it. This is called the dielectric constant, or It is called the relative permittivity,  r Consider later, but Basically e 0 is replaced by the product e 0 e r The capacitance for a parallel plate capacitor becomes Q =  r  0 V 0 A/d and C =  r  0 A/d

26. Real capacitors come in all shapes and sizes - the geometry is more complext than parallel plate, but - the principle is the same