Aim of the lectur e More detail on Capacitance Ohms Law Capacitance Energy storage Dielectrics Ohms Law

1. Aim of the lecture • More detail on • Capacitance • Ohms Law • Capacitance • Energy storage • Dielectrics • Ohms Law • resistance • Power dissipation •   Main learning outcomes • familiarity with • Dielectrics • Resistance • Addition of • Capacitors • Resistors • Power dissipation in resistors Lecture 5

2. + - Dielectrics Material between plates • A dielectric is • an insulator • either polar • or non-polar No dielectric between plates C = e0A/d With a dielectric, then C = ere0A/d

3. +ve -ve Polar Dielectric Electrons leave one plate The same number arrive on the other • In a polar material • molecules are polarised • act like small dipoles • orientate to align with E field E [pure water is like this]

4. ∫ ∫ +ve -ve Polar Dielectric Consider Gauss’ Law surface as shown net charge inside surface reduced charge = e0E.dA E But charge is now q-q’ where q is the charge without dielectric q’ is the charge due to polar molecules q-q’ = e0E.dA

5. ∫ +ve -ve Polar Dielectric q-q’ = e0E.dA E which gives that the electric field is E = (q-q’)/e0A compared with E0 = q/e0A with no field The electric field is weaker when a dielectric is present for the sameapplied voltage The quantity of charge q’ depends on E0, (often) which depends on q so q’ is proportional to q

6. +ve -ve Polar Dielectric so q’ = const q and E = const E0 E define that 1/const is er with the result that C = ere0A/d The quantity of charge q’ depends on E0, (often) which depends on q so q’ is proportional to q

7. Other geometries are possible cylindrical

8. Sphere above a plane

9. A non-polar dielectric is one where the molecules • are non-polar • In this case the molecules are CAUSED to be polar by the electric field, they are • Induced dipoles In practice the only difference is that the values of erare (usually) smaller than for a polar dielectric

10. Add a second capacitor in series Adding Capacitance The net charge on thesetwo plates iszero, it is justan equipotentialline

11. Adding Capacitance Overall effect is to double distance between plates but C = ere0A/d so capacitance is halved

12. C Total capacitance = C/2 C Adding capacitors in series REDUCES the total capacitance

13. 1 1 1 = + CT C1 C2 More generally When adding capacitors in series

14. Example: capacitance with partial dielectric

15. Capacitors Connected in Parallel • Adding a second identical capacitor in parallel • doubles the area of the plates • doubles charge stored for same voltage applied but C = ere0A/d so capacitance is doubled

16. CT C1 C2 = + More generally When adding capacitors in parallel

17. 1 1 1 = + CT C1 C2 CT C1 C2 = + PARALEL SERIES

18. Energy Stored The energy stored is equal to the energy in theelectric field between the plates. Q = CV Work done to move a small charge, dq from one plate to the other is dW = VdQ = VCdV So total energy, E is E = ∫dW = C∫VdV = ½CV2

19. 1 1 1 = + CT C1 C2 CT C1 C2 = + Summary for Capacitors Q = CV E = ½CV2 For parallel addition For series addition C = ere0A/d for parallel plate capacitor e0 is 8.854×10−12 F m–1 er is typically between 1 and 10

20. And this means the maximumvoltage you can put acrossit is 100V Practical Information: Capacitors are labelledin a ‘funny’ way The units are always mF or pF More than you wanted to know! not examinable but useful This means 100k pF ie 100 x 103 x 10-12 F = 100nF (it cant be mF because it wouldbe too big – you just have toknow this!)

21. This little + meansthat this terminalmust be +ve comparedwith the other 22u 35 means 22mF max volts = 35V Final Warning: some capacitors are ‘polarised’ you MUST put the voltage across them the correct direction. Tantalum bead capacitors can explode if connected the wrong way round!!

22. More on Ohms Law and Resistance Recall that V = IR Where V is the voltage applied across a resistance, R and I is the current that flows. The resistance is analogous to the resistance of a pipeto the flow of water through it.

23. Electrons are made to drift in an electric field caused by an external voltage. • They loose energy in collisions with the fixed atoms • They therefore do not accelerate • They drift at constant speed

24. Consider a resistance with a voltage across it. V Suppose the current that flows is Ia Current = 2Ia If we apply the same voltage across two such resistancesconnected in parallel, then the current doubles, so the resistance is inversely proportional to the area, A, of conductor

25. If we put two in series, then we need a voltage V across each to drive the same current, Ia, so resistance is proportional to length, L Resistance = r L/A for many materials r is a constant called the ‘resistivity’ of the material and is very different for different conductors.

26. V = I R • Any real circuit has resistance • Usually wires are a small resistance • we ignore it, assume it is zero • Represent the resistance with a • RESISTOR • The wires that we draw joining parts of a circuit are • Taken to have zero resistance • Resistance is represented by • An ampmeter has zero resistance • A voltmeter has a very high resistance (infinite if perfect)

27. More practical details! Here is how you can tell what a resistor value is: These resistors are 100W with 5% accuracy

28. Resistors in series. Recall that for identical resistors, the resistance is proportionalto the length. This generalises for resistors in series Rtotal = R1 + R2 + R3 + …..

29. For resistors in parallel, it is like increasing the area, so two in parallel gives half the resistance, and in general: Rtotal

30. The energy transferred to the atoms when the electrons collide with them in a resistor is converted to heat Power = current x voltage P = IR but remember ohms law V=IR So P = I2R P = V2/R