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Transcript

PHYS 1442 – Section 001 Lecture #3 Wednesday, June 10, 2009 Dr. Jaehoon Yu • Chapter 16 – • Electric Field Lines Chapter 17 – – – – – Electric Potential Energy Electric Potential Electric Potential and Electric Field Equi-potential Lines The Electron Volt, a Unit of Energy Today’s homework is homework #2, due 9pm, next Thursday!! Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 1 Announcements • Your five extra credit points for e-mail subscription extended till midnight tonight! Please take a full advantage of the opportunity. – Seven of you have subscribed so far. Thank you!!! • All of you have registered in the homework system. – Fantastic job!! – Remember, the due is 9pm tomorrow. • Reading assignments: CH17 – 6 and 17 – 10 • Summer clinic hours extended to cover – Mon – Fri: 11am – 6pm – Saturdays: 11am – 4pm Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 2 Reminder: Special Project – Angels & Demons • Compute the total possible energy released from an annihilation of x-grams of anti-matter and the same quantity of matter, where x is the last two digits of your SS#. (20 points) – Use the famous Einstein’s formula for mass-energy equivalence • Compute the power output of this annihilation when the energy is released in x ns, where x is again the last two digits of your SS#. (10 points) • Compute how many cups of gasoline (8MJ) this energy corresponds to. (5 points) • Compute how many months of electricity usage it corresponds to (3.6GJ). (5 points) • Due by the beginning of the class Monday, June 15. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 3 Reminder: Special Project – Magnitude of Forces • What is the magnitude of the Coulomb force one proton exerts to another 1m away? (10 points) • What is the magnitude of the gravitational force one proton exerts to another 1m away? (10 points) • Which one of the two forces is larger and by how many times? (10 points) • Due at the beginning of the class Monday, June 22. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 4 The Electric Field • Both gravitational and electric forces act over a distance without touching objects What kind of forces are these? – Field forces • Michael Faraday developed an idea of field. – Faraday argued that the electric field extends outward from every charge and permeates through all of space. • Field by a charge or a group of charges can be inspected by placing a small test charge in the vicinity and measuring the force on it. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 5 The Electric Field • The electric field at any point in space is defined as the force exerted on a tiny positive test charge divide by the magnitude of the test charge F F qE – Electric force per unit charge E q • What kind of quantity is the electric field? – Vector quantity. Why? • What is the unit of the electric field? – N/C • What is the magnitude of the electric field at a distance r from a single point charge Q? F 1 Q kQq r 2 kQ 2 E 2 4 r q q r 0 Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 6 Direction of the Electric Field • If there are more than one charge, the individual field due to each charge is added vectorially to obtain the total field at any point. ETot E1 E2 E3 E4 .... • This superposition principle of electric field has been verified by experiments. • For a given electric field E at a given point in space, we can calculate the force F on any charge q, F=qE. – What happens to the direction of the force and the field depending on the sign of the charge q? – The two are in the same directions if q>0 – The two are in opposite directions if q<0 Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 7 Field Lines • The electric field is a vector quantity. Thus, its magnitude can be expressed in the length of the vector and the arrowhead pointing to the direction. • Since the field permeates through the entire space, drawing vector arrows is not an ideal way of expressing the field. • Electric field lines are drawn to indicate the direction of the force due to the given field on a positive test charge. – Number of lines crossing unit area perpendicular to E is proportional to the magnitude of the electric field. – The closer the lines are together, the stronger the electric field in that region. – Start on positive charges and end on negative charges. Earth’s G-field lines Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 8 Electric Fields and Conductors • The electric field inside a conductor is ZERO in the static situation. (If the charge is at rest.) Why? – If there were an electric field within the conductor, there would be a force on free electrons inside of it. – The electrons will move until they reached positions where the electric field becomes zero. – Electric field, however, can exist inside a non-conductor. • Consequences of the above – Any net charge on a conductor distributes itself on the surface. – Although no field exists inside a conductor, the fields can exist outside the conductor due to induced charges on either surface – The electric field is always perpendicular to the surface outside of the conductor. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 9 Example 16-10 • Shielding, and safety in a storm. A hollow metal box is placed between two parallel charged plates. What is the field like in the box? • If the metal box were solid – The free electrons in the box would redistribute themselves along the surface so that the field lines would not penetrate into the metal. • The free electrons do the same in hollow metal boxes just as well as it did in a solid metal box. • Thus a conducting box is an effective device for shielding. Faraday cage • So what do you think will happen if you were inside a car when the car was struck by a lightening? Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 10 Motion of a Charged Particle in an Electric Field • If an object with an electric charge +q is at a point in space where electric field is E, the force exerting on the object is F qE. • What do you think will happen to the charge? – Let’s think about the cases like these on the right. – The object will move along the field line…Which way? – The charge gets accelerated. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 11 Example • Electron accelerated by electric field. An electron (mass m = 9.1x10-31kg) is accelerated in the uniform field E (E=2.0x104N/C) between two parallel charged plates. The separation of the plates is 1.5cm. The electron is accelerated from rest near the negative plate and passes through a tiny hole in the positive plate. (a) With what speed does it leave the hole? (b) Show that the gravitational force can be ignored. Assume the hole is so small that it does not affect the uniform field between the plates. The magnitude of the force on the electron is F=qE and is directed to the right. The equation to solve this problem is F qE ma F qE The magnitude of the electron’s acceleration is a m m Between the plates the field E is uniform, thus the electron undergoes a uniform acceleration eE 1.6 10 C 2.0 10 a m 9.110 kg 19 Wednesday, June e 10, 2009 4 N /C 3.5 10 31 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 15 m s2 12 Example cont’d Since the travel distance is 1.5x10-2m, using one of the kinetic eq. of motions, v2 v02 2ax v 2ax 2 3.5 1015 1.5 102 1.0 107 m s Since there is no electric field outside the conductor, the electron continues moving with this speed after passing through the hole. • (b) Show that the gravitational force can be ignored. Assume the hole is so small that it does not affect the uniform field between the plates. The magnitude of the electric force on the electron is Fe qE eE 1.6 1019 C 2.0 104 N / C 3.2 1015 N The magnitude of the gravitational force on the electron is FG mg 9.8 m s 2 9.1 1031 kg 8.9 1030 N Thus the gravitational force on the electron is negligible compared to the electromagnetic force. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 13 Gauss’ Law • Gauss’ law states the relationship between electric charge and electric field. – More general and elegant form of Coulomb’s law. • The electric field by the distribution of charges can be obtained using Coulomb’s law by summing (or integrating) over the charge distributions. • Gauss’ law, however, gives an additional insight into the nature of electrostatic field and a more general relationship between the charge and the field Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 14 Electric Flux • Let’s imagine a surface of area A through which a uniform electric field E passes • The electric flux is defined as – E=EA, if the field is perpendicular to the surface – E=EAcosθ, if the field makes an angle θ to the surface • So the electric flux is defined as E E A. • How would you define the electric flux in words? – Total number of field lines passing through the unit area perpendicular to the field. N E EA E Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 15 Example of Flux • Electric flux. (a) Calculate the electric flux through the rectangle in the figure (a). The rectangle is 10cm by 20cm and the electric field is uniform with magnitude 200N/C. (b) What is the flux in figure if the angle is 30 degrees? The electric flux is E E A EA cos So when (a) θ=0, we obtain E EA cos EA 200 N / C 0.1 0.2m 2 4.0 N m 2 C And when (b) θ=30 degrees, we obtain 2 2 200 N / C 0.1 0.2 m cos30 3.5 N m C E EA cos30 Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 16 Electric Potential Energy • Concept of energy is very useful solving mechanical problems • Conservation of energy makes solving complex problems easier. • When can the potential energy be defined? – Only for a conservative force. – The work done by a conservative force is independent of the path. What does it only depend on?? • The difference between the initial and final positions – Can you give me an example of a conservative force? • Gravitational force, Spring force • Is the electrostatic force between two charges a conservative force? – Yes. Why? – The dependence of the force to the distance is identical to that of the gravitational force. • The only thing matters is the direct linear distance between the object not the path. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 17 Electric Potential Energy • What does this mean in terms of energies? – The electric force is a conservative force. – Thus, the mechanical energy (K+U) is conserved under this force. – A charged object has only the electric potential energy at the positive plate. – The electric potential energy decreases and turns into kinetic energy of the charge object as the electric force works on the charged objects and the charged object gains speed. • Point of greatest potential energy for – Positively charged object – Negatively charged object Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu PE= U KE= 0 ME= U U+K 0 K K 18 Electric Potential • How is the electric field defined? – Electric force per unit charge: F/q • We can define electric potential (potential) as – The electric potential energy per unit charge – This is like the voltage of a battery… • Electric potential is written with the symbol V – If a positive test charge q has potential energy Ua at a point a, the electric potential of the charge at that point is Ua Va q Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 19 Electric Potential • Since only the difference in potential energy is meaningful, only the potential difference between two points is measurable • What happens when the electric force does “positive work”? – The charge gains kinetic energy – Electric potential energy of the charge decreases • Thus the difference in potential energy is the same as the negative of the work, Wba, done on the charge by the electric field to move the charge from point a to b. • The potential difference Vba is U b U a Wba Vba Vb Va q q PHYS 1442-001, Summer 2009 Dr. – Electric potential is independent of the test charge!! Wednesday, June 10, 2009 Jaehoon Yu 20 A Few Things about Electric Potential • What does the electric potential depend on? – Other charges that creates the field – What about the test charge? • No, the electric potential is independent of the test charge • Test charge gains potential energy by existing in the potential created by other charges • Which plate is at a higher potential? – Positive plate. Why? • Since positive charge has the greatest potential energy on it. – What happens to the positive charge if it is let go? • It moves from higher potential to lower potential – How about a negative charge? • Its potential energy is higher on the negative plate. Thus, it moves from negative plate to positive. Potential difference is the same. Zero point of electric potential unit of the electric potential is Volt (V). can be chosen arbitrarily. • The • From the definition, 1V = 1J/C. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu Often the ground, a conductor 21 connected to Earth, is zero. Example 17 – 1 A negative charge: Suppose a negative charge, such as an electron, is placed at point b in the figure. If the electron is free to move, will its electric potential energy increase or decrease? How will the electric potential change? • An electron placed at point b will move toward the positive plate since it was released at its highest potential energy point. • It will gain kinetic energy as it moves toward left, decreasing its potential energy. • The electron, however, moves from the point b at a lower potential to point a at a higher potential. ΔV=Va-Vb>0. • This is because the potential is generated by the charges on the plates not by the electron. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 22 Electric Potential and Potential Energy • What is the definition of the electric potential? – The potential energy difference per unit charge • OK, then, how would you express the potential energy that a charge q would obtain when it is moved between point a and b with the potential difference Vba? U b U a q Vb Va qVba – In other words, if an object with charge q moves through a potential difference Vba, its potential energy changes by qVba. • So based on this, how differently would you describe the electric potential in words? – A measure of how much energy an electric charge can acquire in a given situation – A measure of how much work a given charge can do. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 23 Comparisons of Potential Energies • Let’s compare gravitational and electric potential energies m • 2m What are the potential energies of the rocks?• – mgh and 2mgh • – QVba and 2QVba Which rock has a bigger potential energy? • – The rock with a larger mass • Why? – It’s got a bigger mass. Wednesday, June 10, What are the potential energies of the charges? Which object has a bigger potential energy? – The object with a larger charge. • Why? – It’s got a bigger charge. PHYS 1442-001, Summer 2009 Dr. 24 The potential is the same but the heavier rockYuor larger charge can do a greater work. 2009 Jaehoon Electric Potential and Potential Energy • The electric potential difference gives potential energy or possibility to do work depending on the charge of the object. • So what is happening in batteries or generators? – They maintain a potential difference. – The actual amount of energy used or transformed depends on how much charge flows. – How much is the potential difference maintained by a car’s battery? • 12Volts – If for a given period, 5C charge flows through the headlight lamp, Joules what is the total energy transformed? • Etot=5C*12V=60 Umm… What is the unit? – If it is left on twice as long? Etot=10C*12V=120J. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 25 Some Typical Voltages Sources Thundercloud to ground Approximate Voltage 108 V High-Voltage Power Lines Power supply for TV tube Automobile ignition 106 V 104 V 104 V Household outlet Automobile battery Flashlight battery Resting potential across nerve membrane 102 V 12 V 1.5 V 10-1 V Potential changes on skin (EKG and EEG) 10-4 V Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 26 Example 17 – 2 Electrons in TV tube: Suppose an electron in the picture tube of a television set is accelerated from rest through a potential difference Vba=+5000V. (a) What is the change in potential energy of the electron? (b) What is the speed of the electron (m=9.1x10-31kg) as a result of this acceleration? (c) Repeat for a proton (m=1.67x10-27kg) that accelerates through a potential difference of Vba=-5000V. • (a) What is the charge of an electron? – e 1.6 1019 C • So what is the change of its potential energy? U qVba eVba 1.6 1019 C 5000V 8.0 1016 J Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 27 Example 17 – 2 • (b) Speed of the electron? – The entire potential energy of the electron turns to its kinetic energy. Thus the equation is 1 K me ve2 0 W U eVba 2 19 1.6 10 ve 2 eVba me C 5000V 8.0 1016 J 2 8.0 1016 7 4.2 10 m/ s 31 9.1 10 • (C) Speed of a proton? 1 K m p v 2p 0 W U e Vba eVba 8.0 1016 J 2 2 8.0 1016 2 eVba 5 vp 9.8 10 m/ s 27 mp 1.67 10 Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 28 Electric Potential and Electric Field • The effect of a charge distribution can be described in terms of electric field or electric potential. – What kind of quantities are the electric field and the electric potential? • Electric Field: Vector • Electric Potential: Scalar – Since electric potential is a scalar quantity, it is often easier to handle. • Well other than the above, what are the connections between these two quantities? Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 29 Electric Potential and Electric Field • The potential energy is expressed in terms of a conservative force Ub U a b a F dl • For the electrical case, we are more interested in the potential difference: Ub U a Vba Vb Va q b a F dl q b a E dl – This formula can be used to determine Vba when the electric field is given. • When the field is uniform Vb Va b a E dl E b a dl Ed or Vba Ed Wednesday, June 10, field in termsPHYS 1442-001, SummerV/m 2009 Dr. Can you derive this from N/C? 30 Unit of the electric of potential? 2009 Jaehoon Yu Example 17 – 3 Uniform electric field obtained from voltage: Two parallel plates are charged to a voltage of 50V. If the separation between the plates is 5.0cm, calculate the magnitude of the electric field between them, ignoring any fringe effect. 5cm 50V What is the relationship between electric field and the potential for a uniform field? V Ed Solving for E Wednesday, June 10, 2009 50V V 50V 1000V / m E 2 d 5.0cm 5 10 m PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 31 Electric Potential due to Point Charges • What is the electric field by a single point charge Q at a distance r? Q 1 Q E 4 0 r k 2 r2 • Electric potential due to the field E for moving from point ra to rb in radial direction away from the charge Q is Vb Va rb ra Q 4 0 Wednesday, June 10, 2009 E dl rb ra Q 4 0 rb ra rˆ ˆ rdr 2 r 1 Q 1 1 dr 2 4 0 rb ra r PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 32 Electric Potential due to Point Charges • Since only the differences in potential have physical meaning, we can choose Vb 0 at rb . • The electrical potential V at a distance r from a single point charge is 1 Q V 4 0 r • So the absolute potential by a single point charge can be thought of as the potential difference by a single point charge between r and infinity Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 33 Properties of the Electric Potential • What are the differences between the electric potential and the electric field? 1 Q – Electric potential V 4 0 r • Electric potential energy per unit charge • Inversely proportional to the distance • Simply add the potential by each of the charges to obtain the total potential from multiple charges, since potential is a scalar quantity 1 Q – Electric field E 2 4 r 0 • Electric force per unit charge • Inversely proportional to the square of the distance • Need vector sums to obtain the total field from multiple charges • Potential for the positive charge is large near the charge and decreases towards 0 at a large distance. • Potential for the negative charge is large negative near the Wednesday, June 10, PHYS 1442-001, Summer 2009 Dr. 34 2009 Jaehoon 0 Yu at a large distance. charge and increases towards Shape of the Electric Potential • So, how does the electric potential look like as a function of distance? – What is the formula for the potential by a single charge? 1 Q V 4 0 r Positive Charge Negative Charge Uniformly charged sphere would have the potential the same as a single point charge. Wednesday, June 10, PHYS 1442-001, Summer 2009 Dr. 35 What does this mean? Uniformly charged sphere behaves 2009 Jaehoon Yulike all the charge is on the single point in the center. Example 23 – 6 Work to bring two positive charges close together: What minimum work is required by an external force to bring a charge q=3.00μC from a great distance away (r=infinity) to a point 0.500m from a charge Q=20.0 μC? What is the work done by the electric field in terms of potential energy and potential? q Q Q W qVba 4 0 rb ra Since rb 0.500m, ra we obtain q Q 8.99 109 N m2 C 2 3.00 106 C 20.00 106 C q Q W 1.08J 0 0.500 m 4 0 rb 4 r 0 b Electric force does negative work. In other words, the external force must work +1.08J to bring the charge 3.00mC from infinity to 0.500m to the charge 20.0mC. Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 36 Electric Potential by Charge Distributions • Let’s consider that there are n individual point charges in a given space and V=0 at r=infinity. • Then the potential due to the charge Qi at a point a, Qi 1 distance ria from Qi is Via 4 0 ria • Thus the total potential Va by all n point charges is n n Qi 1 Via Va i 1 4 0 ria i 1 • For a continuous charge distribution, we obtain Wednesday, June 10, 2009 V PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 1 4 0 dq r 37 Example 23 – 8 • Potential due to a ring of charge: A thin circular ring of radius R carries a uniformly distributed charge Q. Determine the electric potential at a point P on the axis of the ring a distance x from its center. • Each point on the ring is at the same distance from the point P. What is the distance? r R2 x2 • So the potential at P is 1 dq 1 What’s this? V dq 4 0 r 4 0 r Q 1 dq 2 2 2 2 4 0 x R 4 0 x R Wednesday, June 10, 2009 PHYS 1442-001, Summer 2009 Dr. Jaehoon Yu 38