1. Monday Feb 17. 2014 PHYS 1442-004, Dr. Andrew Brandt 1 PHYS 1442 – Section 004 Lecture #10+Review Monday February 17 2014 Dr. Andrew Brandt CH 18 Alternating Current Short review for test on ch 16-19

2. 2 PHYS 1442-004, Dr. Andrew Brandt Alternating Current Does the direction of the flow of current change when a battery is connected to a circuit? –No. Why? Because its source of potential difference is constant. –This kind of current is called the Direct Current (DC How would DC look as a function of time? –A horizontal line Electric generators at electric power plant produce alternating current (AC) –AC reverses direction many times a second –AC is sinusoidal as a function of time Most currents supplied to homes and business are AC. Monday Feb 17. 2014

3. 3 PHYS 1442-004, Dr. Andrew Brandt Alternating Current The voltage produced by an AC electric generator is sinusoidal –This is why the current is sinusoidal Voltage produced can be written as What are the maximum and minimum voltages? V0 V0 and –V 0  The potential oscillates between +V 0 and –V 0, the peak voltages or amplitude  What is f ? The frequency, the number of complete oscillations made per second. What is the unit of f ? What is the normal size of f in the US? –f = 60 Hz in the US and Canada. –Many European countries have f = 50Hz.   f

4. Monday Feb 17. 2014 4 PHYS 1442-004, Dr. Andrew Brandt Alternating Current Since V=IR, if a voltage V exists across a resistance R, the current I is What are the maximum and minimum currents? –I0 –I0 and –I 0 –The current oscillates between +I 0 and –I 0, the peak currents or amplitude. The current is positive when electron flows in one direction and negative when they flow in the opposite direction. – What is the average current? Zero. So there is no power and no heat produced in a heater? –Wrong! The electrons actually flow back and forth, so power is delivered. What is this?

5. Monday Feb 17. 2014 5 PHYS 1442-004, Dr. Andrew Brandt Power Delivered by Alternating Current AC power delivered to a resistance is: –Since the current is squared, the power is always positive The average power delivered is Since the power is also P=V 2 /R, we can obtain The average of the square of current and voltage are important in calculating power: Average power

6. Monday Feb 17. 2014 6 PHYS 1442-004, Dr. Andrew Brandt Power Delivered by Alternating Current The square root of each of these are called root-mean-square, or rms: rms values are sometimes called effective values –These are useful quantities since they can substitute current and voltage directly in power equations, as if they were DC values –In other words, an AC of peak voltage V0 V0 or peak current I0 I0 produces as much power as DC voltage of V rms or DC current I rms. –So normally, rms values in AC are specified or measured. US uses 115 to 120 V rms voltage. What is the peak voltage? Europe uses 240V

7. Monday Feb 17. 2014 7 PHYS 1442-004, Dr. Andrew Brandt How does one feel an electric shock? –Electric current stimulates nerves and muscles, and we feel a shock –The severity of the shock depends on the amount of current, how long it acts and through what part of the body it passes –Electric current heats tissues and can cause burns Currents above 70mA on a torso for a second or more is fatal, causing heart to function irregularly, “ventricular fibrillation” Dry skin has a resistance of 10 4 to 10 6 .. When wet, it could be 10 3 . A person in good contact with the ground who touches 120V DC line with wet hands can receive a fatal current Electric Hazards: Leakage Currents

8. Monday Feb 17. 2014 8 PHYS 1442-004, Dr. Andrew Brandt At temperatures near absolute 0K, the resistivity of certain materials approaches 0. –This state is called the “superconducting” state. –Observed in 1911 by H. K. Onnes when he cooled mercury to 4.2K (-269 o C). Resistance of mercury suddenly dropped to 0. –In general superconducting materials become superconducting below a transition temperature. –The highest temperature superconductor so far is 160K First observation above the boiling temperature of liquid nitrogen is in 1987 at 90K observed from a compound of yttrium, barium, copper and oxygen. Since a much smaller amount of material can carry just as much current more efficiently, superconductivity can make electric cars more practical, computers faster, and capacitors store higher energy (not to mention LHC magnets) Superconductivity

9. Quick Review Test Weds on CH 16-19 Multiple choice, bring a 50 problem Scantron form Monday Feb 17. 2014 9 PHYS 1442-004, Dr. Andrew Brandt

10. Monday Feb 17. 2014 10 Electric Charge and Conservation Two types of electric charge –Like charges repel while unlike charges attract The net amount of electric charge produced in any process is ZERO!! When a positively charged metal object is brought close to an uncharged metal object –If the objects touch each other, the free charges flow until an equilibrium state is reached (charges flow in a conductor.) –If the objects are close, the free electrons in the neutral object still move within the metal toward the charged object leaving the opposite end of the object positively charged.(induced charge) PHYS 1442-004, Dr. Andrew Brandt

11. Monday Feb 17. 2014 11 Coulomb’s Law – The Formula A vector quantity. Newtons Direction of electric (Coulomb) force (Newtons) is always along the line joining the two objects. Unit of charge is called Coulomb, C, in SI. Elementary charge, the smallest charge, is that of an electron: -e where Formula PHYS 1442-004, Dr. Andrew Brandt

12. Monday Feb 17. 2014 12 Vector Problems Calculate magnitude of vectors (Ex. force using Coulomb’s Law) Split vectors into x and y components and add these separately, using diagram to help determine sign Calculate magnitude of resultant |F|=  (F x 2 +F y 2 ) Use  = tan -1 (F y /F x ) to get angle PHYS 1442-004, Dr. Andrew Brandt

13. Monday Feb 17. 2014 13 PHYS 1442-004, Dr. Andrew Brandt After calculating magnitudes, take x+y components and then get total force Angle:

14. Monday Feb 17. 2014 14 The Electric Field The electric field at any point in space is defined as the force exerted on a tiny positive test charge divided by magnitude of the test charge The electric field inside a conductor is ZERO in a static situation PHYS 1442-004, Dr. Andrew Brandt

15. Example Calculate the total electric field (a) at point A and (b) at point B in the figure due to both charges, Q 1 and Q 2. Solution: The geometry is shown in the figure. For each point, the process is: calculate the magnitude of the electric field due to each charge; calculate the x and y components of each field; add the components; recombine to give the total field. a. E = 4.5 x 10 6 N/C, 76° above the x axis. b. E = 3.6 x 10 6 N/C, along the x axis. 15 Monday Feb 17. 2014PHYS 1442-004, Dr. Andrew Brandt

16. Monday Feb 17. 2014 16 Example Electron accelerated by electric field. An electron (mass m = 9.1x10 -31 kg) is accelerated from rest in a uniform field E (E = 2.0x10 4 N/C) between two parallel charged plates (d=1.5 cm), andpasses through a tiny hole in the positive plate. (a) With what speed does it leave the hole? PHYS 1442-004, Dr. Andrew Brandt

17. Monday Feb 17. 2014 17 Electric Potential Energy Concept of energy is very useful solving mechanical problems Conservation of energy makes solving complex problems easier. Defined for conservative forces (independent of path) PHYS 1442-004, Dr. Andrew Brandt

18. Monday Feb 17. 2014 18 Electric Potential Energy What is the definition of change in electric potential energy U b –U a ? – The potential gained by the charge as it moves from point a to point b. – The negative work done on the charge by the electric force to move it from a to b. Parallel plates w/ equal but opposite charges –The field between the plates is uniform since the gap is small and the plates are infinitely long… What happens when we place a small charge, +q, on a point at the positive plate and let go? –The electric force will accelerate the charge toward negative plate and it gains kinetic energy PHYS 1442-004, Dr. Andrew Brandt

19. Monday Feb 17. 2014 19 Electric Potential The electric field (E) is defined as electric force per unit charge: F/q (vector quantity) Electric potential (V) is defined as electrical potential energy per unit charge U/q (scalar) PHYS 1442-004, Dr. Andrew Brandt

20. Monday Feb 17. 2014 20 Comparisons of Potential Energies Let’s compare gravitational and electric potential energies 2mm What are the potential energies of the rocks? –mgh and 2mgh Which rock has a bigger potential energy? –The rock with a larger mass Why? –It’s got a bigger mass. What are the potential energies of the charges? –+QV ba and +2QV ba Which object has a bigger potential energy? –The object with a larger charge. Why? –It’s got a bigger charge. The “potential” is the same but the heavier rock or larger charge can do a greater work. PHYS 1442-004, Dr. Andrew Brandt

21. Monday Feb 17. 2014 21 What are the differences between the electric potential and the electric field? –Electric potential (U/q) Simply add the potential from each of the charges to obtain the total potential from multiple charges, since potential is a scalar quantity –Electric field (F/q) Need vector sums to obtain the total field from multiple charges Potential for a positive charge is large near a positive charge and decreases to 0 at large distances. Potential for the negative charge is small (large magnitude but negative) near the charge and increases with distance to 0 Properties of the Electric Potential PHYS 1442-004, Dr. Andrew Brandt

22. Monday Feb 17. 2014 22 Electrostatic Potential Energy; Three Charges Work is needed to bring all three charges together –Work needed to bring Q 1 to a certain place without the presence of any charge is 0. –Work needed to bring Q 2 to a distance to Q 1 is –Work need to bring Q 3 to a distance to Q 1 and Q 2 is So the total electrostatic potential of the three charge system is PHYS 1442-004, Dr. Andrew Brandt

23. Monday Feb 17. 2014 23 Capacitors A simple capacitor consists of a pair of parallel plates of area A separated by a distance d. –A cylindrical capacitors are essentially parallel plates wrapped around as a cylinder. Circuit Diagram PHYS 1442-004, Dr. Andrew Brandt

24. Monday Feb 17. 2014 24 If a battery is connected to a capacitor, the capacitor gets charged quickly, one plate positive and the other negative with an equal amount.. For a given capacitor, the amount of charge stored in the capacitor is proportional to the potential difference V ba between the plates. C is a proportionality constant, called capacitance of the device. Should know how to add them, calculate based on geometry Capacitors C is a property of a capacitor so does not depend on Q or V. PHYS 1442-004, Dr. Andrew Brandt

25. Monday Feb 17. 2014 25 Dielectrics Capacitors generally have an insulating sheet of material, called a dielectric, between the plates to –Increase the breakdown voltage above that in air –Allows the plates get closer together without touching Increases capacitance ( recall C=  0 A/d) –Also increases the capacitance by the dielectric constant Where C 0 is the intrinsic capacitance when the gap is vacuum, and K or  is the dielectric constant PHYS 1442-004, Dr. Andrew Brandt

26. Monday Feb 17. 2014 26 Electric Current Electric Current: Any flow of charge –Current can flow whenever there is potential difference between the ends of a conductor –Electric current in a wire can be defined as the net amount of charge that passes through a wire’s full cross section at any point per unit time –Average current is defined as: –Current is a scalar –Current is flow of charge, charge is conserved, so current in equals current out at a given point on circuit C/s 1A=1C/s PHYS 1442-004, Dr. Andrew Brandt

27. Monday Feb 17. 2014 27 Ohm’s Law: Resistance The exact amount of current flow in a wire depends on –The voltage –The resistance of the wire to the flow of electrons The higher the resistance the less the current for the given potential difference V PHYS 1442-004, Dr. Andrew Brandt

28. Monday Feb 17. 2014 28 Resistivity It is experimentally found that the resistance R of a metal wire is directly proportional to its length l and inversely proportional to its cross-sectional area A –The proportionality constant  is called the resistivity and depends on the material used. The higher the resistivity the higher the resistance –The reciprocal of the resistivity is called the conductivity, , A l PHYS 1442-004, Dr. Andrew Brandt

29. Monday Feb 17. 2014 29 Electric Power Power -the rate at which work is done or the energy is transferred P=IV can apply to any devices while the formulae involving resistance only apply to Ohmic resistors. PHYS 1442-004, Dr. Andrew Brandt I 2 R used for heat loss Temperature dependence

30. A quiz Monday Feb 17. 2014 30 PHYS 1442-004, Dr. Andrew Brandt

31. Monday Feb 17. 2014 31 PHYS 1442-004, Dr. Andrew Brandt Example Will a 30A fuse blow? Determine the total current drawn by all the devices in the circuit in the figure. The total current is the sum of current drawn by the individual devices. Solve for I BulbHeater StereoDryer Total current What is the total power?

32. Monday Feb 17. 2014 32 PHYS 1442-004, Dr. Andrew Brandt Magnetism So are magnet poles analogous to electric charge? –No. Why not? –While the electric charges (positive and negative) can be isolated, magnet poles cannot. –So what happens when a magnet is cut? You get two magnets! The more they get cut, the more magnets are made –Single-pole magnets are called “monopoles,” but to date none have been observed… Ferromagnetic materials: Materials that show strong magnetic effects –Iron, cobalt, nickel, gadolinium and certain alloys Other materials show very weak magnetic effects

33. Electron charge: Electron mass: Proton mass: Colomb’s Law: : For a point charge: m e = 9.1x10 -31 kg m p =1.67x10 -27 kg V=Ed (uniform field) K.E.=mv 2 /2 Ugrav=mgh g=9.8 m/s 2 Uelec=qV Q=CV C=capacitance parallel plate: dielectric: Cap. stored energy: Ohm’s Law: V=IR Power: P=IV Current: I=q/t AC: Resistivity: Eqs. of motion: 1442 Test I Eq. Sheet |F|=  (F x 2 +F y 2 )  = tan - 1 (F y /F x ) C eq =C 1 +C 2 (parallel) 1/C eq =1/C 1 +1/C 2 (series) Monday Feb 17. 2014 33 PHYS 1442-004, Dr. Andrew Brandt

34. Monday Feb 17. 2014 34 PHYS 1442-004, Dr. Andrew Brandt –The direction of the magnetic field is tangent to a line at any point –The direction of the field is the direction the north pole of a compass would point to –The number of lines per unit area is proportional to the strength of the magnetic field –Magnetic field lines continue inside the magnet –Since magnets always have both poles, magnetic field lines form closed loops, unlike electric field lines Magnetic Field Just like an electric field surrounds electric charge, a magnetic field surrounds a magnet What does this mean? –Magnetic force is also a field force –The force one magnet exerts on another can be viewed as the interaction between the magnet and the magnetic field produced by the other magnet –What kind of quantity is the magnetic field? Vector or Scalar? So one can draw magnetic field lines, too. Vector

35. Monday Feb 17. 2014 35 PHYS 1442-004, Dr. Andrew Brandt Earth’s Magnetic Field Which way does a compass point? So the magnetic pole of the geographic North pole is … –Yep South! –Since the magnetic north pole points to the geographic north, the geographic north must have magnetic south pole The pole in the north is still called geomagnetic north pole just because it is in the north –Similarly, south pole has magnetic north pole To add confusion: the Earth’s magnetic poles do not coincide with the geographic poles  magnetic declination –Geomagnetic north pole is in northern Canada, some 1300km off the true north pole Earth’s magnetic field line is not tangent to the earth’s surface at all points –The angle the Earth’s field makes to the horizontal line is called the angle of dip

36. Monday Feb 17. 2014 36 PHYS 1442-004, Dr. Andrew Brandt Electric Current and Magnetism In 1820, Oersted found that when a compass needle is placed near an electric wire, the needle deflects as soon as the wire is connected to a battery and the current flows –Electric current produces a magnetic field The first indication that electricity and magnetism are linked –What about a stationary electric charge and magnet? They don’t affect each other The magnetic field lines produced by a current in a straight wire is in the form of circles following the “right-hand” rule –The field lines follow right-hand’s fingers wrapped around the wire when the thumb points to the direction of the electric current

37. Monday Feb 17. 2014 37 PHYS 1442-004, Dr. Andrew Brandt Directions in a Circular Wire? OK, then what are the directions of the magnetic fields generated by the current flowing through circular loops?

38. Monday Feb 17. 2014 38 PHYS 1442-004, Dr. Andrew Brandt Magnetic Forces on Electric Current Since electric current exerts force on a magnet, the magnet should also exert force on the electric current –How do we know this? Newton’s 3 rd law (confirmed in this case by Oerste) Direction of the force is always –perpendicular to the direction of the current and also –perpendicular to the direction of the magnetic field, B – How is this possible? Experimentally the direction of the force is given by another right-hand rule  When the fingers of the right-hand point in the direction of the current and the finger tips bend in the direction of magnetic field B, the direction of thumb points to the direction of the force

39. Monday Feb 17. 2014 39 PHYS 1442-004, Dr. Andrew Brandt Another Version of RHR Suppose we keep thumb in I direction, and fingers in B direction, then the palm will give direction of force! Example: Simply evaluate the force on 2 nd wire due to magnetic field in wire 1 The magnetic field at wire 2 from wire 1 is into the page, so the force on wire 2 is to the left. What about the force on wire 1 due to wire 2? What if I flipped the direction of the current on one wire?

40. Monday Feb 17. 2014 40 PHYS 1442-004, Dr. Andrew Brandt Magnetic Forces on Electric Current The right-hand rule gives the direction, but what about magnitude? Observations show that the magnitude is directly proportional to –The current in the wire –The length of the wire in the magnetic field (if the field is uniform) –The strength of the magnetic field The force also depends on the angle  between the directions of the current and the magnetic field –When the wire is perpendicular to the field, the force is the strongest –When the wire is parallel to the field, there is no force at all Thus the force on current I in a wire of length l in a uniform field B is

41. 41 PHYS 1442-004, Dr. Andrew Brandt Magnetic Forces on Electric Current Magnetic field strength B can be defined using the previous proportionality relationship with the constant=1: if  =90 o, and if  =0 o So the magnitude of the magnetic field B can be defined as where F max is the magnitude of the force on a straight length l of wire carrying a current I when the wire is perpendicular to B The relationship between F, B and I can be written in a vector formula: where l is a vector with magnitude is the length of the wire and the direction is that of the conventional current This formula works if B is uniform. If B is not uniform or l does not form the same angle with B everywhere, the infinitesimal force acting on a differential length d l is Monday Feb 17. 2014

42. 42 PHYS 1442-004, Dr. Andrew Brandt About the Magnetic Field, B The magnetic field is a vector quantity The SI unit for B is tesla (T) –What is the definition of 1 Tesla in terms of other known units? –1T=1N/A·m –An older unit: 1 tesla is the same as a weber per meter-squared 1Wb/m 2 =1T The cgs unit for B is gauss (G) –How many T is one G? 1G=10 -4 T –For computation in SI units, one MUST convert Gauss to Tesla Magnetic field on the Earth’s surface is about 0.5G=0.5x10 -4 T On a diagram, for field out of page and for field in.

43. Monday Feb 17. 2014 43 PHYS 1442-004, Dr. Andrew Brandt Example 27 – 1 Measuring a magnetic field. A rectangular loop of wire hangs vertically as shown in the figure. A magnetic field B is directed horizontally perpendicular to the wire, and points out of the page. The magnetic field B is very nearly uniform along the horizontal portion of wire ab (length l =10.0cm) which is near the center of a large magnet producing the field. The top portion of the wire loop is free of the field. The loop hangs from a balance which measures a downward force ( in addition to the gravitational force) of F=3.48x10 -2 N when the wire carries a current I=0.245A. What is the magnitude of the magnetic field B at the center of the magnet? Magnetic force exerted on the wire due to the uniform field is SinceMagnitude of the force is Solving for B What happened to the forces on the loop on the side? The two forces cancel out since they are in opposite direction with the same magnitude. What about gravitational force?

44. Monday Feb 17. 2014 44 PHYS 1442-004, Dr. Andrew Brandt Example 27 – 2 Magnetic force on a semi-circular wire. A rigid wire, carrying a current I, consists of a semicircle of radius R and two straight portions as shown in the figure. The wire lies in a plane perpendicular to the uniform magnetic field B 0. The straight portions each have length l within the field. Determine the net force on the wire due to the magnetic field B 0. As in the previous example, the forces on the straight sections of the wire are equal and opposite direction. Thus they cancel. Since What is the net X component of the force acting on the circular section? Integrate over    What do we use to figure out the net force on the semicircle? We divide the semicircle into infinitesimal straight sections. 0 Why? Because the forces on left and the right-hand sides of the semicircle balance. Y-component of the force dF is dF y = dFsinø = IB 0 Rdøsinø Which direction? Vertically upward direction. The wire will be pulled further into the field.