# Review. Spring Semester /21/14. Physics for Scientists & Engineers 2 1

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1 Review Spring Semester 2014 Physics for Scientists & Engineers 2 1

2 Notes! Homework set 13 extended to Tuesday, 4/22! Remember to fill out SIRS form: Physics for Scientists & Engineers 2 2

3 Homework hints I! Particle track detectors are used to measure the speed of particles if the lifetime of the particle is known. Particle X has a lifetime of ps. These particles are created in an experiment inside the detector by a given reaction. The particles leave 13.2 cm long tracks on average before they decay into other particles not observable by the detector. What is the average speed of the particles in terms of the speed of light?! Answer: relativistic particle! Lifetime is always given in restframe of the particle! Lifetime in detector frame: t = γt 0! Distance: l = v t = vγt 0! Solve for v Physics for Scientists & Engineers 2 3

4 Homework hints II! How much energy is required to accelerate a spaceship with a rest mass of 113 metric tons to a speed of c?! Work = kinetic energy K = (γ-1)mc E21 J! Every day our Earth receives J energy from the Sun. If we were able to use 0.85 percent of this energy to accelerate spaceships, then how many missions would be possible in one year?! 1.55E22 J per day is 5.66E24 J per year! 0.85% is of J = 4.81E22 J = 48.1E21 J! Number of missions: 48.1/1.89 = 25.4! 25 complete missions are possible.! For more hints, see Physics for Scientists & Engineers 2 4

5 PHY 184 Final Exam! When: Wednesday, April 30, 8pm to 10pm! Where: BPS 1410 (this room)! You can bring two pages of equation sheets: Two pieces of letter-size paper Handwritten! Also bring: No. 2 pencil Calculator Student ID! The exam is closed book and will cover the entire semester! Assigned seating! A practice final exam has been uploaded to the lecture notes folder Physics for Scientists & Engineers 2 5

6 Review - Electric charge! There are two types of charge: negative and positive.! Most objects are electrically neutral; they have equal numbers of negative and positive charges (net charge is 0).! An object becomes charged by adding or removing electrons.! An electron carries negative charge of magnitude e = C.! Law of Charges Like charges repel and opposite charges attract.! Law of charge conservation The total charge of an isolated system is strictly conserved.! Conductors are materials where some of the electrons can move freely.! Insulators are materials where none of the charges can move freely. Physics for Scientists & Engineers 2 6

7 Review - Coulomb s Law! The electric force F between two charges, q 1 and q 2, separated by a distance r is given by Coulomb s Law: F = qq k r 1 2 2! 1/r 2 dependence Opposite charges: F is attractive (-) Like charges: F is repulsive (+)! The constant k is called Coulomb s constant and is given by Nm 1 C k = or k = where ε = C 4πε 0 Nm Physics for Scientists & Engineers 2 7

8 The Electric Field! A charge creates an electric field around itself and the other charge feels that field. + + Test charge q Electric field at a given point in space: place a positive test charge q at the point and measure the electrostatic force that acts on the test charge; then E = F q Test charge: point object with a very small positive charge so that it does not modify the original field Physics for Scientists & Engineers 2 8

9 Review -Field Lines from a Point Charge! The electric field lines from a point charge extend out radially.! For a positive point charge, the field lines point outward Terminate at infinity! For a negative charge, the field lines point inward Originate at infinity E = k q r 2 ˆr Physics for Scientists & Engineers 2 9

10 Review - Gauss Law! Gauss Law ε 0 Φ = q! Gauss Law says that the electric flux through a closed surface is proportional to the net charge enclosed by this surface. Flux Φ = E d A! If we add the definition of the electric flux we get another expression for Gauss Law ε 0 E d A = q Physics for Scientists & Engineers 2 10

11 Review - Electric Fields from Charge Distributions! The electric field E at distance r from a wire with charge density λ is E λ = = 2πε r 0 2kλ r! The electric field inside a closed conductor is 0! Electric field lines are always perpendicular to the surface of a conductor Physics for Scientists & Engineers 2 13

12 Review - Electric Fields from Charge Distributions (2)! The electric field E produced by an infinite conducting plane with charge density σ is E = σ ε 0! The electric field E produced by an infinite non-conducting plate with charge density σ is E = σ 2ε 0 Physics for Scientists & Engineers 2 14

13 Review: Electric Field (3)! The electric field inside a spherical shell of charge q is zero! The electric field outside a spherical shell of charge q is the same as the field from a point charge q E = 1 4πε! Electric field from charge distributed uniformly throughout a sphere of radius r 0 q r 2 r 2 > r r 1 < r E r E r 2 = k q t r 2 2 ( ) = qr = kqr ( ) t 1 t πε 0r r Physics for Scientists & Engineers 2 15

14 Review - Electric Potential! Taking the convention that the electric potential is zero at infinity we can express the electric potential in terms of the electric field as! Calculate the electric field from gradients of the electric potential in each component direction V x ; V E = Ey = ; E V z = x y z! The electric potential from a point charge q at a distance r is given by: V! The electric potential can be expressed as an algebraic sum of all sources of electric potential V = kq r n = i= 1 V i In particular for a system of point charges: V kq n n i V i i= 1 i= 1 ri = = Physics for Scientists & Engineers 2 16

15 Force Field Energy - Potential kqq/ R 2 Integrate over distance Differentiate with respect to distance kqq/r Divide by q Multiply with q Divide by q Multiply with q kq/r 2 Point charges Integrate over distance Differentiate with respect to distance kq/r

16 Review: Capacitance (1)! The definition of capacitance is q C = V! The capacitance of a parallel plate capacitor is given by A is the area of each plate d is the distance between the plates! The capacitance of a spherical capacitor is C = ε 0 A d C 4πε r 1 is the radius of the inner sphere r 2 is the radius of the outer sphere = 0 rr 1 2 r r 2 1 Physics for Scientists & Engineers 2 19

17 Review: Capacitance (2)! The capacitance of an isolated spherical conductor is C = 4πε R! The capacitance of a cylindrical capacitor is q 0 λl 2πε L 0 C = = = V λ ln 2/ r r 1 2πε 0 ( r r ln ) ( 2/ 1)! Placing a dielectric between the plates of a capacitor increase the capacitance by κ C = κc air! The electric potential energy stored in a capacitor is given by U = 1 2 CV 2 Physics for Scientists & Engineers 2 20

18 Review: Capacitors in parallel and in series! The equivalent capacitance for n capacitors in parallel is C eq n = C i= 1 i! The equivalent capacitance for n capacitors in series is C n 1 1 eq = i= 1 C = i = Physics for Scientists & Engineers 2 21

19 Review: Current! Electric current i is the net charge passing a given point in a given time i = dq dt! The ampere is abbreviated as A and is given by 1 C 1 A = 1 s! The current per unit area flowing through a conductor is the current density J! If the current is constant and perpendicular to a surface, then and we can write an expression for the magnitude of the current density J = i A Physics for Scientists & Engineers 2 22

20 Review - Ohm s law! The property of a particular device or object that describes it ability to conduct electric currents is called the resistance, R! The definition of resistance R is! The unit of resistance is the ohm, Ω R = V i 1 Ω= 1 V 1 A Physics for Scientists & Engineers 2 23

21 Review - Resistance! The resistance R of a device is given by R L = ρ A! ρ is resistivity of the material from which the device is constructed! L is the length of the device! A is the cross sectional area of the device Physics for Scientists & Engineers 2 24

22 Review Parallel and Series Resistors! We can replace n parallel resistors with one equivalent resistor given by n 1 1 = R eq i= 1 R i! We can replace n series resistors with one equivalent resistor given by R eq n = i= 1 R i Physics for Scientists & Engineers 2 25

23 Power, Kirchhoff s Law! Power dissipated in a circuit P= iv = i R= 2 V! To analyze any circuit, identify loops and junctions At each junction, define a direction for the current Making sure the current directions are consistent across all junctions For each loop, define an analysis direction! Kirchhoff s Junction Rule The sum of the currents entering a junction must equal the sum of the currents leaving a junction! Kirchhoff s Loop Rule The sum of voltage drops around a complete circuit loop must sum to zero. 2 R i 1 V 1 i 2 i 3 V 2 V 3 Physics for Scientists & Engineers 2 26

24 Power dissipated by a resistor in a network! Four resistors, R 1 = 17.6Ω, R 2 = 35.8Ω, R 3 = 95.6Ω and R 4 = 21.0Ω are connected to a 18.0 V battery as shown in the figure. Determine the power dissipated by resistor R Find the equivalent resistance. 2. Go backwards and find the current through R Calculate the power. Correct answer: 1.56W Physics for Scientists & Engineers 2 27

25 ! Charging a capacitor:! Discharging a capacitor: q = q 0 e t RC! Time constant: i Review: RC Circuit t qt () = q0 1 e τ t V emf RC dq = = e dt R i = dq dt = q 0 RC e τ = RC t RC Physics for Scientists & Engineers 2 28

26 Problem solving strategy! Use this approach to solve problems, in particular if at first you have no clue. Step 1 recognize Step 2 describe Step 3 plan Step 4 execute Step 5 evaluate Recognize the problem What s going on? Think Describe the problem in terms of the field What does this have to do with? Sketch Plan a solution How do I get out of this? Research Execute the plan Let s get an answer! Simplify, Calculate, Round Evaluate the solution Can this be true? Double-check! Draw a picture! Phrase the question in your own words! Relate the question to something you just learned! Identify physics quantities, forces, fields, potentials,! Find a physics principle (symmetry, conservation, )! Write down the equations! Solve equations, starting with intermediate steps! Check units, order-of-magnitude, insert into original question, Physics for Scientists & Engineers 2 30

27 Practice question: Three charges on a line! Three +20 µc charges are placed along a straight line, successive charges being 2m apart as shown in the figure. What is the force on the charge on the right end. A) N B) N C) N D) N q 1 q 2 q 3 2m 2m Physics for Scientists & Engineers 2 31

28 Practice question: Three charges on a line! Three +20 μc charges are placed along a straight line, successive charges being 2m apart as shown in the figure. What is the force on the charge on the right end. B) N q 1 q 2 q 3 2m 2m Physics for Scientists & Engineers 2 32

29 Practice question: Resistor Circuit! In the circuit to the right, what is the current I when the voltage between points a and b is 12.0V? (R 1 =47 Ω, R 2 =120 Ω, R 3 =270 Ω) A) I=0.14A B) I=0.40A C) I=0.09A D) I=0.03A Physics for Scientists & Engineers 2 33

30 Practice question: Resistor Circuit! In the circuit to the right, what is the current I when the voltage between points a and b is 12.0V? (R 1 =47 Ω, R 2 =120 Ω, R 3 =270 Ω) C) I=0.09A R 2 and R 3 are in parallel: R 1 and R 23 are in series: = + R23 = 83.08Ω R R R R = R + R R = Ω Current is given by I = V/R = A Physics for Scientists & Engineers 2 34

31 Practice question: Charge on a capacitor! A certain parallel plate capacitor consists of two plates, each with area 200cm 2, separated by a 0.4cm air gap. The capacitor is connected to a 500V power supply. If a liquid with κ=2.6 is poured between the plates, how much additional charge will flow onto the capacitor? A) 35.2 nc B) 70.1 nc C) no additional charge D) 9.0 μc Physics for Scientists & Engineers 2 35

32 Practice question: Charge on a capacitor! A certain parallel plate capacitor consists of two plates, each with area 200cm 2, separated by a 0.4cm air gap. The capacitor is connected to a 500V power supply. If a liquid with κ=2.6 is poured between the plates, how much additional charge will flow onto the capacitor? A) 35.2 nc Physics for Scientists & Engineers 2 36

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