Physics 121: Electricity and Magnetism Introduction

1. Physics 121: Electricity and Magnetism Introduction • Syllabus, rules, assignments, exams, etc. • Text: Young & Friedman, University Physics • Homework & Tutorial System: Mastering Physics • 5 Weeks: Stationary charges – • Forces, fields, Electric flux, Gauss’ Law, • Potential, potential energy, capacitance • 2 Weeks: Moving charges – • Currents, resistance, circuits containing resistance and capacitance, • Kirchoff’s Laws, multi-loop circuits, RC circuits • 2 Weeks: Magnetic fields (static fields due to moving charges) • Magnetic force on moving charges, • Magnetic fields caused by currents (Biot-Savart’s and Ampere’e Laws) • 2 Weeks: Induction & Inductance • Changing magnetic flux (field) produces currents (Faraday’s Law) • Inductance, LR Circuits • 2 – 3 Weeks: AC (LCR) circuits, • Eelectromagnetic oscillations, Resonance • Impedance, Phasors • Not covered: • Maxwell’s Equations - unity of electromagnetism • Electromagnetic Waves – light, radio, gamma rays,etc Course Content:

2. Physics 121 - Electricity and MagnetismLecture 01/02 – Vectors, Charge, Coulombs Law Y&F Chapter 21, Sec. 1 - 3 • Review of Vectors: • Components in 3D. Right-hand rule • Scalar multiplication, Dot product, vector product • Charge: • Basic properties of Electric Charge • Quantization of Charge • Structure of Matter • Conservation of Charge • Conductors and Insulators • Coulomb’s Law – Force • Shell Theorem • Examples

3. z y x Definition: Right-Handed Coordinate Systems • We always use right-handed coordinate systems. • In three-dimensions the right-hand rule determines which way the positive axes point. • Curl the fingers of your RIGHT HAND so they go from x to y. Your thumb will point in the positive z direction. This course uses several right hand rules related to this one!

4. z az q ay ax f y x a sin(q) Vectors in 3 dimensions Rene Descartes 1596 - 1650 • Unit vector (Cartesian) notation: • Spherical polar coordinate representation: 1 magnitude and 2 directions • Conversion into x, y, z components • Conversion from x, y, z components

5. z q r y f x Angle range for a full sphere: Factors into 2 simple 1 dimensional integrations Surface Integral Example: Show that the surface area of a sphere A= 4pR2 by integrating over the sphere’s surface Find dA – an area segment on the surface of the sphere, then integrate on angles f (azimuth) and q (co-latitude). • Where: • dl is a curved length segment of the circle around • the z-axis (along a constant latitude line) • dh is a segment along the q direction (along a • constant longitude line)

6. Multiplication of a vector by a scalar: vector times scalar  vector whose length is multiplied by the scalar Dot product (or Scalar product or Inner product): - vector times vector  scalar - projection of A on B or B on A - commutative unit vectors measure perpendicularity: f There are 3 Kinds of Vector Multiplication

7. Cross product (or Vector product or Outer product): • Vector times vector  another vector perpendicular to the • plane of A and B • - Draw A & B tail to tail: right hand rule shows direction of C Ø f • If A and B are parallel or the same, A x B = 0 • If A and B are perpendicular, A x B = AB (max) Algebra: Unit vector representation: i j k Applications: Vector multiplication, continued

8. Fields are used to explain “Action at a Distance” • Place a test mass, test charge, or test current at some test point in a field • It feels a force due to the presence of remote sources of the field. • The sources “alter space” at every possible test point. • The forces (vectors) at a test point due to multiple sources add up • via superposition (the individual field vectors add & form the net field). See also: Fields Examples – Gravitation posted on web page

9. GENERATING STATIC CHARGE • Feet on nylon carpet, dry weather  sparks • Face of monitor or TV  dust, sizzling if touched • Rub glass rod (+) with silk (-)  • Rub plastic rod (-) with fur (+)  CHARGED ROD ATTRACTS NEUTRAL CONDUCTOR BY INDUCTION. + + + + + + + + + + + + + + + + + + - - - - - - - - + + + + + + F - F CHARGES WITH SAME SIGN REPEL F - - - + + + copper - F CHARGES WITH OPPOSITE SIGN ATTRACT - F F WHAT’S CHARGE? WEAK CHARGE IS A BASIC ATTRIBUTE OF MATTER, LIKE MASS • mass = source of gravitational field strength of response to gravitational field • charge = source of electrostatic (electromagnetic) field strength of response to electromagnetic field STRONG

10. … BUT…matter is normally electrically neutral (equal numbers of + and – charges). (Why?) OBJECTS WITH A NET CHARGE HAVE LOST OR GAINED SOME BASIC CHARGES • Net + charge  deficit of electrons (excess of “holes”) • Net – charge  excess electrons (deficit of “holes”). • Why isn’t all matter a gas? Electrical forces hold matter together. • Outside neutral objects electrical forces are “screened”, but not completely • Weak “leftover” electrical force (Van der Waal’s) makes atoms stick together • & defines properties of solids, liquids, and gases. • CHARGE IS A “CONSERVED” QUANTITY • LIKE ENERGY, MOMENTUM, ANGULAR MOMENTUM, usually MASS • Net charge never just disappears or appears from nowhere. • Charges can be cancelled or screened by those of opposite sign • All of chemistry is about exchanging & conserving charge among atoms & molecules • Nuclear reactions conserve charge, e.g. radioactive decay: 92U238  90Th234 + 2He4 • Pair Production & annihilation: g  e- + e+ (electron – positron pair) ALL MATTER IS ELECTRICAL…

11. Electron Proton 2-1: Which type of charge is easiest to pull out of an atom? • Proton • Electron The protons all repel each other Why don’t nuclei all fly apart? Neutron The physical world we see is electrical • MODERN ATOMIC PHYSICS (1900 – 1932): Rutherford, Bohr • Charge is quantized: e = 1.6x10-19 coulombs (small). • Every charge is an exact multiple of +/- e. Milliken ~1900. • Electrons ( -e) are in stable orbital clouds with radius ~ 10-10 m., • mass me = 9.1x10-31 kg. (small). Atoms are mostly space. • Protons ( +e) are in a small nucleus, r ~10-15 m , mp= 1.67x10-27 kg, • Ze = nuclear charge, Z = atomic number. There are neutrons too. • A neutral atom has equal # of electrons and protons. • An ion is charged + or – and will attract opposite charges to become neutral

12. What’s the Difference between Insulators & Conductors? Insulators Charge not free to move (polarization not shown) induced charge induced charge net charge Conductor Charge free to move in copper rod Plastic rod repels electrons Charge separation induced in cooper rod F always attractive for plastic rod near either end of copper rod What if plastic rod touches copper rod?

13. Charging a conducting sphere by induction - - - - + + + + + + + + + + + + + + + + + + + + + + + + - - - - ground wire - - - - 5. + + + uniform distribution Atomic view of Conductors: charges free to move 3. 1. 2. + + + - + + - + SOLID CONDUCTORS • Regular lattice of fixed + ions • conduction band free electrons • wander when E field is applied + + + - draw off negative charge neutral induce charges metals for example LIQUID & GASEOUS CONDUCTORS • Electrons and ions both free • to move independently • Normally random motion • Electrons & ions move in opposite • directions in an E field. - - - - - - - - - 4. - - - - - - - - - - - - - sea water, humans, hot plasmas - - - - remove wire - - - -

14. free charge free charge Polarization means charge separates but does not leave home - - - - + + + + - - - - + + + + MOLECULES CAN HAVE PERMANENT POLARIZATION... ...OR MAY BE INDUCED TO ROTATE OR DISTORT WHEN CHARGE IS NEAR positive polarization charge field inside insulating material is smaller than it would be in vacuum negative polarization charge - + THE DIELECTRIC CONSTANT MEASURES MATERIALS’ ABILITY TO POLARIZE Semiconductors: - Normally insulators, but can be weak conductors when voltage, doping, or heat is applied - Bands are formed by overlapping atomic energy levels conduction band ENERGY BAND GAP valence band Insulators: Charges not free to move Electrons are tightly bound to ions... ...BUT insulators can be induced to polarize by nearby charges Insulators can be solids, liquids, or gases

15. 3rd law pair of forces F12 q1 r12 q2 F21 Gravitation is weak G m m 1 2 Unit of charge = Coulomb 1 Coulomb = charge passing through a cross section of a wire carrying 1 Ampere of current in 1 second 1 Coulomb = 6.24x1018 electrons [ = 1/e] = F 12 2 r 12 » - G . x 11 N . m 2 / kg 2 6 67 10 ELECTROSTATIC FORCE LAW (coulomb, 1785) Constant k=8.89x109 Nm2/coul2 Force on q1 due to q2 (magnitude) repulsion shown • Force is between pairs of point charges • An electric field transmits the force • (no contact, action at a distance) • Symmetric in q1 & q2 so F12 = - F21 • Inverse square law • Electrical forces are strong compared to gravitation • e0= 8.85 x 10-12Why this value? Units

16. Force on q1 due to q2 y q1 Displacement to q1fromq2 Unit vector pointing radially away from q2 at location of q1 q2 x squared cubed Coulomb’s Law using vector notation Sketch shows repulsion: For attraction (opposite charges):

17. y +q2 +q3 x +q1 -q4 Superposition of forces & fields The electrostatic force between a specific pair of point charges does not depend on interaction with other charges that may be nearby – there are no 3-body forces (same as gravitation) Example: Find NET force on q1 Method: find forces for all pairs of charges involving q1, then add the forces vectorially • F11 is meaningless • Use coulomb’s law to calculate individual forces • Keep track of direction, usually using unit vectors • Find the vector sum of individual forces at the point • For continuous charge distributions, integrate instead of summing

18. -2q +Q -q Example: Charges in a Line 2-4: Where do I have to place the +Q charge in order for the forces on it to balance, in the figure at right? • Cannot tell, because + charge value is not given. • Exactly in the middle between the two negative charges. • On the line between the two negative charges, but closer to the -2q charge. • On the line between the two negative charges, but closer to the -q charge. • There is no location that will balance the forces.

19. -2q +Q L y -q Calculate the Exact Location for the forces to balance • Force on test charge Q is attractive toward both negative charges, hence they could cancel. • By symmetry, position for +Q in equilibrium is along y-axis • Coordinate system: call the total distance L and call y the position of charge +Q from charge -q. • Net force is sum of the two force vectors, and has to be zero, so • k, q, and Q all cancel due to zero on the right, so our answer does not depend on knowing the charge values. We end up with • Solving for y, , y is less than half-way to the top negative charge

20. 1 proton in nucleus, neutral Hydrogen atom has 1 electron • Both particles have charge e = 1.6 x 10-19 C. • me = electron mass = 9.1 x 10-31 kg. • mp = proton mass = 1833 me • a0 = radius of electron orbit = 10-8 cm = 10-10 m. • Electrostatics dominates atomic structure by a factor of ~ 1039 • So why is gravitation important at all? Matter is normally Neutral! • Why don’t atomic nuclei with several protons break apart? How do we know that electrostatic force - not gravitational force - holds atoms & molecules together? Example: Find the ratio of forces in Hydrogen

21. 3 1 2 r23 r12 p e p F12 FBD for 1 F13 F21 FBD for 2 F23 F32 F31 FBD for 3 Example: Forces on an electron and two protons along a line (1D) By symmetry, all forces lie along the horizontal axis Show forces on each of the objects in free body diagrams for # 1, 2, & 3 • There are 3 action-reaction pairs of forces • Which forces are attractive/repulsive? • Could net force on any individual charge be zero above? • What if r12 = r23?

22. F12 F21 R x q1 q2 EXAMPLE: Vector-based solution… … Start with two point charges on x-axis Find F12 - the Force on q1 due to q2 F21 has the same magnitude, opposite direction Let: q1 = +e = +1.6 x 10-19 C q2 = +2e = +3.2 x 10-19 C. R = 0.02 m Magnitude of force: small! In vector notation:

23. R F12 F21 r23 r13 x q3 q1 q2 R F12 F13 q1 r13 F21 F31 F32 q3 r23 F23 …see next page... q2 EXAMPLE continued: add a 3rd charge on x-axis between q1 and q2 Let: q3 = - 2e = -3.2 x 10-19 C. r13= ¾ R, r23= ¼ R What changes? • Because force is pairwise: F12 & F21 not affected...but... • Four new forces appear, only two distinct new magnitudes • F13 and F23 .... FBDs below.

24. EXAMPLE continued: calculate magnitudes Find net forces on each of the three charges by applying superposition On 1: On 2: On 3:

25. a a a a a Another Example: Find the final charges after the spheres below touch? B B B +20e -15e What determines how much charge ends up on each? -30e A A A -15e -50e • Example: Movement of Charge in Conductors • Two identicalconducting spheres. Radii small compared to a. • Sphere A starts with charge +Q, sphere B is neutral • Connect them by a wire. What happens and why? Charge redistributes until ......??? How do we know the final charges are equal? Are they equal if the spheres are not identical? B B B +Q/2 +Q/2 A A A +Q +Q/2 +Q/2 • Spheres repel each other. How big is the force? • We applied shell theorem outside • Do the spheres approximate point • charges? What about polarization?

26. Proof by lengthy integration or symmetry + Gauss Law + + + + + + q r + + P + + R + + + + + + + Outside a shell, r > R: Uniform shell responds to charged particle q as if all the shell’s charge is a point charge at the center of the shell. Inside a shell, r < R: Charged particle q inside uniform hollow shell of charge feels zero net electrostatic force from the shell. Why does the shell theorem work? spherical symmetry  cancellations What force does a fixed spherical charge distribution exert on a point test charge nearby? • Shell Theorem (Similar to gravitational Shell Theorem) • Holds for spherical symmetry only: shell or solid sphere • Uniform surface charge density s - not free to move(insulator) • Shell has radius R, total charge Q

27. q Q q Q 2R d d • The test charge induces polarization charge on the sphere • Charge distribution will NOT be uniform any more • Extra surface charge is induced on the near and far surfaces of the sphere • The net force will be slightly more attractive than for point charges The Shell Theorem works approximately for conductors if the test charge induces very little polarization; e.g., if q << Q and/or d >> R. Charge on a spherical conductor – charge free to move • Net charge Q on a spherical shell spreads out everywhere over the surface. • Tiny free charges inside the conductor push on the others and move as far apart as they can go. When do they stop moving? If no other charges are nearby (“isolated” system) • Charges would spread themselves out uniformly because of symmetry • Sphere would act as if all of it’s charge is concentrated at the center (for points outside the sphere) as in the Shell Theorem. Bring test charge q<<Q near the sphere Q is a point charge here

28. y qprot q1 q2 Region III Region I Region II Both forces to the right No equilibrium possible Proton in Region I: qprot Proton in Region III: qprot Forces opposed but and Forces can never balance Proton in Region II: qprot Forces opposed and So balance point exists here L x Equilibrium x: Did the test charge magnitude make any difference? Example: three charges along x-axis q1 = + 8q q not stated q2 = -2q qproton = +e For q1 & q2 as in the sketch where should a proton (test charge) be placed to be in equilibrium (Fnet = 0, use “free body diagrams” & symmetry)