1. Physics 7C, Lecture 4 Winter Quarter -- 2007 Fields, Forces, Potentials, and Energy Professor Robin Erbacher 343 Phy/Geo erbacher@physics.ucdavis.edu

2. Announcements Course syllabus (policy, philosophy) on the web: http://physics7.ucdavis.edu Unit 9 begins today: DLMs 8 -14. No quiz today, see calendar on web. Quiz #2 rubric to be posted soon. Quizzes every other lecture, ~20 minutes each, average of 4 best = 45% (or 20)% of grade. Turn off cell phones and pagers during lecture.

3. Force Fields: Not just history or sci-fi! Gravity, Electricity, Magnetism… Fields, they are… Yes! If the Jedi path you follow, understand them, you must. Gravity, Electricity, Magnetism… Fields, they are… Yes! If the Jedi path you follow, understand them, you must.

4. Force Models In Physics 7B you learned about contact forces: normal and friction, gravitational, electric. We will call this the Direct Model of Forces It’s straightforward to think about a ball bouncing off the ground due to direct contact with the ground. But: How does Earth exert its gravitational force on the ball while in mid-air? This is an example of action-at-a-distance, and leads to Field Model of Forces Object A Object B exerts force field Object B exerts force Object A field creates

5. Scalar Field: Temperature v. position The temperature at every location on the globe.  T(x,y) x=longitude y=latitude T axis is color. For every position in space (x,y) we associate a quantity (Temperature, in this case). Since the quantity is a scalar, this is a scalar field. Same color = same temperature, lines of “equal temperature” www.wunderground.com

6. Vector Field: Velocity field of viscous fluid flow in a pipe Velocity of fluid at a distance R from the center of the pipe has both magnitude and direction Velocity Field We associate an arrow (vector) to each point in our space of interest (in this case, inside the pipe).

7. MAP?  spatial variation (x,y,z) * Temperature Scalars:* Elevation * Atmospheric Pressure Scalar = magnitude only * Wind Velocity Vectors: * Gravity Field g = GM/r 2 * Electric Field E = kQ/r 2 Vector = magnitude and direction ex: velocity v {3 components… (v x,v y,v z ) } Recall: What is a Field? What is a Field? => a “map” of a measurable quantity Side out in Bold means vector or overstrike arrow: v Quantities?

8. Gravity Field Maps What is a Field? => a “map” of a measurable quantity Notice that the magnitude of the vectors increase for larger mass M: strength of field is greater!

9. Vector Addition Side out in Recall your vector addition rules: Whether we are discussing force vectors or field vectors, the rules of vector addition are simple. Always add vectors head-to-toe. Then it doesn’t matter what order you add them in. The length of a vector is in general proportional to the magnitude. The magnitude of a vector is a scalar: a simple numerical quantity. You can break it down into x and y components to add the vectors. ˆ ˆ

10. Gravitational Fields and Forces For gravity, we can think about an Object with mass M exerting a force on another object with mass m. Alternatively, we can think about an Object with mass M creating a gravitational field. This field would then act on any other object nearby, such as one with mass m. What does g depend on? What units does it have? In which direction does it point?

11. Alternatively, we can think about a charge Q creating an electric field. This field would then act on any other charges nearby, such as one with charge q. Electric Fields and Forces For electricity, we can use the direct force model similarly to gravity. Consider a charge Q exerting a force on a new charge q: What does E depend on?What units? In which direction does it point?

12. For an electric field E : Magnitude: Direction: out from +Q in toward -Q Electric Field/Force Directions For the coulomb force on a test charge q in a field E : Magnitude: Direction: along the E field vector for +q opposite the E field vector for -q q Q r

13. Deduce the sign of the charges Two charges, q 1 and q 2, have equal magnitudes q and are placed as shown in the figure above. The net electric field at point P is vertically upward. What can we conclude about the charges q 1 and q 2 from the resulting electric field? Principle of superposition. PRS Question: a)q 1 positive, q 2 negative b)q 1 negative, q 2 positive c)q 1 & q 2 have the same sign

14. Electric Field Lines This is an Electric Dipole! Like charges (++) Opposite charges (+ -)

15. Electric Field Strengths Typical electric field strengths: 1 cm away from 1 nC of negative charge E = kq /r 2 = 10 10 * 10 -9 / 10 -4 =10 5 N /C Note: (N*m 2 /C 2 ) C / m 2 = N/C Fair weather atmospheric electricity = 100 N/C downward at 100 km high in the ionosphere Field due to a proton at the location of the electron in the H atom. (The radius of the electron orbit is 0.5*10 -10 m) E = kq /r 2 = 10 10 * 1.6*10 -19 / (0.5 *10 -10 ) 2 = 4*10 11 N /C q r E - - - - - - - - - E +++++ + Hydrogen atom 1 N / C = Volt / meter

16. Example: Calculating E Fields Finding an electric field from two charges: We have q 1 = +10 nC at the origin, q 2 = +15 nC at x=4 m. What is E at y=3 m and x=0? (point P) x y q 1 =10 nC q 2 =15 nC 4 3 P Use principle of superposition. ( Find x and y components of electric field due to both charges and add them up.)

17. Example: Calculating E Fields Recall: E =kq/r 2 x q 1 =10 nC q 2 =15 nC 4 5  y 3 E  Field due to q 1 : E = 10 10 N.m 2 /C 2 10 X10 -9 C/(3m) 2 = 11 N/C in the y direction. E y = 11 N/C E x = 0 Field due to q 2 : 10 10 N.m 2 /C 2 15 x10 -9 C/(5m) 2 = 6 N/C at some angle  Resolve into x and y components. E y = E sin   C E x = E cos   C Now add all components: E y = 11 + 3.6 = 14.6 N/C E x = -4.8 N/C Magnitude :    atan E y /E x = atan (14.6/-4.8)= 72.8 deg

18. Electric Field Lines The picture above is a 3-D visualization of the electric field between two oppositely charged plates. The electromagnetic force is one of the 4 fundamental forces of nature, by moving electrons around, we can do calculations inside computers. When the fields are strong enough to move a LOT of electrons through the air, we see lightning as a result.

19. Charge Induction Inducing Charge on a Net Neutral Object: How can a neutral object create an Electric field? (Where would the charges come from to produce such a field?) Static Electricity: Charge can be transferred from one object to another by rubbing. Static is the imbalance of positive and negative charges.

20. Van de Graaff generator

21. Field model of gravitational forces g field Mass m The gravitational field is a map of vectors at each and every point in space. Each vector represents the direction and magnitude of the gravitational force per mass (that’s g). F on m = mg, pointing in the direction g. Mass M   field g   force on mass m

22. Escaping gravity… Is it possible to go far enough away to be beyond the Earth’s gravitational field? a)Yep, and you should go there b)No, gravity will always weigh you down c)It depends on the mass of the gravitational source d)It depends on the mass of the object in the field PRS Question: Just as a light bulb spreads out waves of light everywhere, as the light particles travel outward for ever, gravity's influence is everywhere. You are matter and have your own gravitationally field spreading to the ends of space. Your influence is everywhere…truly! 4 x area from 2 x distance

23. Gradient Relations: Potential Energy Recall: What is the potential energy of a mass m in a the Earth’s gravitational field, a height h above the surface of the Earth? PE = mgh ! Force on a mass m in gravity field g is F = mg. Magnitude of force is the spatial derivative, or gradient, of the potential energy of the mass: The direction of the force on the mass m is toward decreasing PE grav (hence the negative sign!) Gradient relation

24. Gravitational Potenial Energy Force increases with greater slope Potential Energy difference r - 0 PE = GMm/r F = -  PE /  r, the - slope Negative means rapid decrease of PE with decreasing r

25. Field model of electrical forces The source charges somehow alter the space around them. This is the electric field that is created. The test charges in the field experience a force from the field E field from a -Q source -Q Charge -Q   field E   force on charge q q

26. Electric Forces and Potential Energy Electric potential energy is the gradient of the force (change of force): depends on position, and distances. +- E +q Does potential energy increase or decrease if the charge +q moves from the positive plate to the negative plate? We need more information to get the value of the PE change: distance between the plates, voltage drop across the plates… PRS Question: 1)Increases 2)Decreases 3) Need more info

27. Gradients for E Forces: Potential Energy Force on a charge q in an Electric field E is F = qE. Magnitude of force is the spatial derivative, or gradient, of the potential energy of the charge: The direction of the force on the charge +/- q is toward decreasing PE charge (hence the negative sign again!)

28. Gradients for E Fields: Electric Potential Slope of the Electric Potential Constant with distance Negative Electric field is Constant as a function of distance Positive Introducing Electric Potential V (voltage)

29. Electric Potential V (volts) Electric potential V depends on position, and distances. The electric field E can be determined by the spatial derivative of an electric potential, V. Equipotential surfaces for point charge: Lines where V is the same. Circles are 0.5 volts apart, but distances between are NOT uniform. Circles get closer and closer toward center. Potential grows as 1/r.

30. Electric potential of point charges We lose kinectic energy as we get closer, until we stop and rebound Positive charge: potential hillNegative charge: potential well We gain kinectic energy as we get closer, it pulls us in!