Announcements

1. Announcements Today Selected Review of older material Even the full notes from today (online) are not comprehensive Some Basic definitions Maxwell’s Equations Circuits Final Comprehensive!!! Special emphasis last few lectures 12 Multiple Choice 5 Short Answer 1 Problem (chosen from a few) hard!

2. Coulomb’s Law • Like charges repel, unlike charges attract • Force is directly along a line joining the two charges q1 q2 r ke= 8.988109 Nm2/C2 • An inverse square law, just like gravity • Can be attractive or repulsive – unlike gravity • Constant is enormouscompared to gravity

3. Quiz Two test charges are brought separately into the vicinity of a charge +Q. First, test charge +q is brought to point A a distance r from charge +Q. Next, the +q charge is removed and a test charge +2q is brought to point B a distance 2r from charge +Q. Compared with the electric field of the charge at A, the electric field of the charge at B is: • T +Q +Q +q +2q A B • Greater • Smaller • The same.

4. + + - - Electric Field Lines • Graphical Illustration of Electrical Fields • Lines start on positive charges and end on negative • Number of lines from/to a charge is proportional to that charge • Density of lines tells strength of field.

5. Electric Field Lines Consider the four field patterns below: Assuming that there are no charges in the region of space depicted, which field pattern(s) could represent electrostatic field(s)?

6. P Magnetic Field B Current I Currents make Magnetic Fields Magnetic Field B Current I • Magnetic Fields are created by electric currents • They are perpendicular to both the direction of the current and the separation between the wire and the point • Falls off with distance from wire • Direction of magnetic field is given by right-hand rule

7. ds The Biot-Savart Law • Must add up, i.e. integrate, the contribution from the entire current distribution Magnetic Field B Current I

8. Fields we know and love: Wires • Infinite straight wire B=m0I(r)/(2pR2) • Inside the wire • Finite Straight wire

9. Fields we know and love: Other • A single circular loop (Current Coil), has an axial field • A solenoid has a uniform field inside • A plane of current has a uniform field

10. Inductance and Inducers • Unit is Vs/A also called a henry (H) • Circuit diagram looks like this: • Any loop of wire (especially a coil) will have a certain reluctance to change its current • This reluctance is called the inductance • Denoted by the letter L Length l N turns R Current I

11. Capacitors • Capacitors accumulate separated charge • Capacitance is measure of how well a capacitor stores charge. • Farad (F) is C/V, measures capacitance • Capacitors also store energy • Typical capacitances measured in mF,F or pF • The capacitance depends on the geometry of the capacitor and the • presence or absence of a dielectric medium .

12. Resistance • Define resistance as the ratio of the voltage to the current • Resistance is measured in units of Ohms () • Resistance is always positive • Current always flows from positive to negative • Note: This is not Ohm’s law! We can (in principle) always use this Electric Field E Area A Current I Length L

13. – + Kirchoff’s Rules • The total current flowing into a point must equal the total current flowing out of a point [conservation of charge] • The total voltage change around a loop must total zero I2 V1 V2 I3 I1 V3 I3=I2+I1 V1 + V2 + V3 = 0

14. Using Kirchoff’s Rules • Draw a circuit diagram and label everything known or unknown! • To every series of components, assign a direction to the current I (don’t worry if you get it wrong, the result will be correct just negative) • You must be consistent however after you assign a direction! • Write down conservation of charge at each vertex • Write down one equation for each loop • Solve all equations • You might end up with many equations, but I trust that you can solve simultaneous equations.

15. Note on Kirchoff’s Rules • For complex circuits one obtains differential equations • Z.B. RC,LC, RL (RLC not discussed) circuits • Brute force! Will always work (sans mistakes) • You can make life easier by • using obvious parallel and series relations • Using thoughfully picked loops • Remember the Res-monster maze in class • (Ch28 Circuits p653)

17. Example Find the equivalent capacitance of the combination in the figure below. Assume that C1 = 4 µF, C2 = 4 µF, and C3 = 2  µF. • What is the equivalent capacitance of the whole system? • 10 mF C) 4 mF • 1 mF D) 2.5 mF

18. In Series: C1 C2 C3 Series Circuits • Two or more resistors connected together can be treated as one giant resistor • For resistors in series, the current is the same through both of them R1 R2 V1 = R1I V2 = R2I V = V1+V2 = R1I + R2I = (R1 + R2 )I R = R1 + R2 Charges are the same on each capacitor

19. In Parallel: C1 C2 C3 Parallel Circuits • Two or more resistors connected together can be treated as one giant resistor • For resistors in parallel, the voltage is the same across both of them V = R2I2 V = R1I1 R2 R1 Voltages are the same also across each capacitor Same Voltages

20. I • Let I be the current in the circuit RL - Circuits • What happens when the switch Sis closed at t = 0? R + L E – • Use Kirchoffs rule for loops on the circuit S Does this look familiar?

21. RC–Circuits vs RL-Circuits

22. RC–Circuits vs RL-Circuits • At t=0, ordinary wire • At t=0, broken wire, little current • for small t • As t-> infinity, broken wire • As t-> infinity, ordinary wire

23. The diagrams show three circuits with identical batteries, identical inductors, and identical resistors. Just after the switch is closed which has the least current through the battery? The diagrams show three circuits with identical batteries, identical inductors, and identical resistors. Just after the switch is closed which has the greatest current through the battery? The diagrams show three circuits with identical batteries, identical inductors, and identical resistors. A very long time later, which has the least current through the battery? Quiz • What if we had capacitors instead of inductors? Think about this yourself.

24. – + LC - Circuits I • Switch S1 is closed, then opened. • At t = 0, switch S2 is closed. • What happens? C E L S2 S1

25. – + LC - Circuits C E L S2 S1 These equations

26. Electric Potential Point A • Path you choose does not matter. • (conservative) Electric Field E Point B • Factor out the charge – then you have electric potential V

27. Potential difference in constant field + Equipotential surfaces - • Electric Fields are perpendicular to equipotential surfaces • Positively charged particles are attracted to low potential areas (and vice versa) Electric Field High Potential Low Potential

28. Binding Energy and mass • The mass of a nuclei is less than the mass of its parts. • There must exist a binding energy, i.e., an energy keeping the nuclei together. Remember the energy is from electrostatic and nuclear forces! • The binding energy is so large that there is a measurable mass difference. (Unlike binding due to other forces!) • Eb(MeV)=(Zmp+Nmn-MA) x 931.494MeV/amu • Binding energy per nucleon is the binding energy divided by the mass number

29. Maxwell’s Equations Differential Form Integral Form

30. Maxwell and Lorentz Force Law Differential Form

31. Gauss’s Law charge q • Flux out of an enclosed region depends only on total charge inside A positive charge q is set down outside a sphere. Qualitatively, what is the total electric flux out of the sphere as a consequence? A) Positive B) Negative C) Zero D) It is impossible to tell from the given information

32. Gauss’s Law II • Magnetic field lines are always loops, never start or end on anything (no magnetic monopoles) • Net flux in or out of a region is zero • Gauss’s Law for electic fields

33. Ampere’s Law Generalized • When there is a net current flowing into a region, the charge in the region must be changing, as must the electric field. • By Gauss’s Law, the electric flux must be changing as well • Change in electric flux creates magnetic fields, just like currents do • Displacement current – proportional to time derivative of electric flux Magnetic fields are created both by currents carried by conductors (conduction currents) and by time-varying electric fields.

34. Consider three wires with current flowing in/out as shown • Consider three different loops surrounding the wires X 2 A Y 3 A Z 1 A Quiz: Ampere’s Law Which of the loops has the largest and smallest integrals of the magnetic field around the loops drawn? A) X > Y > Z C) Y > Z > X B) X > Z > Y D) Y > X > Z