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Potential at a Certain Location

Potential at a Certain Location. q 2. r 2. q 1. A. r 1. 2. Travel along a path from point very far away to the location of interest and add up at each step:. q 2. q 1. dl. A. E. 1. Add up the contribution of all point charges at this point. Common Pitfall.

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Potential at a Certain Location

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  1. Potential at a Certain Location q2 r2 q1 A r1 2. Travel along a path from point very far away to the location of interest and add up at each step: q2 q1 dl A E 1. Add up the contribution of all point charges at this point

  2. Common Pitfall Assume that the potential V at a location is defined by the electric field at this location. A Example: E = 0 inside a charged metal sphere, but V is not!

  3. A negative test charge Q = -0.6C was moved from point A to point B In a uniform electric field E=5N/C. The test charge is atrestbefore and after the move. The distance between A and B is 0.5m and the line connecting A and B is perpendicular to the electric field. How much work was done by the net external force while moving the test charge from A to B? A E = 5 N/C 0.5m B • 1.5J • 0J • –1.5J • 3.0J • –3.0J

  4. After moving the -0.6C test charge from A to B, it was then moved from B to C along the electric field line. The test charge is at restbefore and after the move. The distance between B and C also is 0.5m. How much work was done by the net external force while moving the test charge from A to C? E = 5 N/C 0.5m C B • 1.5J • 0J • –1.5J • 3.0J • -3.0J A

  5. Instead of moving the test charge from A to B then to C, it is moved from A to D and then back to C. The test charge is at restbefore and after the move. How much work was done by the net external force while moving the test charge this time? A D E = 5 N/C 0.5m 0.5m B C • 1.5J • 0J • –1.5J • Infinitely big • Do not know at this time.

  6. +1.5 J of work was done by the net external force while moving the -0.6 C test charge from B to C. The test charge is at restbefore and after the move. What is the voltage difference between B and C, and at which point is the voltage larger? A D E = 5 N/C 0.5m 0.5m B C • 2.5 V, voltage higher at B. • 2.5 V, voltage higher at C. • 0.9 V, voltage higher at B. • 1.5 V, voltage higher at B. • 1.5 V, voltage higher at C.

  7. Potential Energy and Field Energy In a multiparticle system we can either consider a change in potential energy or a change in field energy (but not both); the quantities are equal. The idea of energy stored in fields is a general one: Magnetic and gravitational fields can also carry energy. The concept of energy stored in the field is very useful: - electromagnetic waves

  8. An Electron and a Positron System Surroundings e- e+ Release electron and positron – the electron (system) will gain kinetic energy Conservation of energy  surrounding energy must decrease Does the energy of the positron decrease? - No, it increases Where is the decrease of the energy in the surroundings? - Energy stored in the fields must decrease

  9. An Electron and a Positron System Surroundings e- e+ Energy: Single charge: Dipole: (far) Energy stored in the E fields decreases as e+ and e- get closer!

  10. An Electron and a Positron System Surroundings e- e+ Principle of conservation of energy: (Field energy) + Kpositron + Kelectron = 0 (Field energy) = -2(Kelectron ) Alternative way: e+and e- are both in the system: Uel = -2(Kelectron ) Change in potential energy for the two-particle system is the same as the change in the field energy

  11. Chapter 18 Magnetic Field

  12. Magnetic Field A compass needle turns and points in a particular direction there is something which interacts with it Magnetic field (B): whatever it is that is detected by a compass Compass: similar to electric dipole

  13. Electron Current Magnetic fields are produced by moving charges Current in a wire: convenient source of magnetic field Static equilibrium: net motion of electrons is zero Can make electric circuit with continuous motion of electrons The electron current (i) is the number of electrons per second that enter a section of a conductor. Counting electrons: complicated Indirect methods: measure magnetic field measure heating effect Both are proportional to the electron current

  14. Exercise If 1.81016 electrons enter a light bulb in 3 ms – what is the magnitude of electron current at that point in the circuit?

  15. Detecting Magnetic Fields We will use a magnetic compass as a detector of B. How can we be sure that it does not simply respond to electric fields? Compass needle: Interacts with iron, steel – even if they are neutral Unaffected by aluminum, plastic etc., though charged tapes interact with these materials Points toward North pole – electric dipole does not do that

  16. The Magnetic Effects of Currents Make electric circuit: What is the effect on the compass needle? What if we switch polarity? What if we run wire under compass? What if we change the current or there is no current in the wire?

  17. The Magnetic Effects of Currents Hans Christian Ørsted (1777 - 1851) Experimental results: • The magnitude of B depends on the amount of current • A wire with no current produces no B • B is perpendicular to the direction of current • B under the wire is opposite to B over the wire Oersted effect: discovered in 1820 by H. Ch. Ørsted How does the field around a wire look?

  18. Magnetic Field Due to Long Current-Carrying Wire

  19. The Magnetic Effects of Currents Principle of superposition: The moving electrons in a wire create a magnetic field What can you say about the magnitudes of BEarthand Bwire? What if BEarth were much larger than Bwire?

  20. Exercise A current-carrying wire is oriented N-S and laid on top of a compass. The compass needle points 27o west. What is the magnitude and direction of the magnetic field created by the wire Bwire if the magnetic field of Earth is BEarth= 210-5 T (tesla).

  21. Biot-Savart Law for a Single Charge Jean-Baptiste Biot (1774-1862) Felix Savart (1791-1841) Nikola Tesla (1856-1943) Electric field of a point charge: Moving charge makes a curly magnetic field: B units: T (tesla) = kg s-2A-1

  22. The Cross Product Calculate magnitude: Calculate direction: Right-hand rule

  23. Question A ) +x W h a t i s t he d ir e ct i o n o f B ) – x C ) +y D ) – y < 0 , 0 , 3 > x < 0 , 4 , 0 > ? E ) z er o m ag n it u d e

  24. Two-dimensional Projections B B B B B • a vector (arrow) is facing into the screen  a vector (arrow) is facing out of the screen r v Why must the field change direction above and below the dashed line?

  25. Exercise What is B straight ahead? What if the charge is negative?

  26. Distance Dependence B1 B2 B3 r v Which is larger, B1or B3? Which is larger, B1or B2?

  27. Moving Charge Sign Dependence r v B1 B1 r v r v B + Magnetic field depends on qv:Positive and negative charges produce the same B if moving in opposite directions at the same speed - For the purpose of predicting B we can describe current flow in terms of ‘conventional current’ – positive moving charges. -

  28. Question An electron passing through the origin is traveling at a constant velocity in the negative y direction. What is the direction of the magnetic field at a point on the positive z axis? y -x +x -z +z No magnetic field x v z

  29. Exercise A current-carrying wire lies on top of a compass. What is the direction of the electron current in this wire? What would the direction of conventional current have to be?

  30. Frame of Reference Any magnetic field? charged tape Electric fields: produced by charges Magnetic fields: produced by moving charges

  31. Frame of Reference Must use the velocities of the charges as you observe them in your reference frame! There is a deep connection between electric field and magnetic fields (Einstein’s special theory of relativity)

  32. Retardation If we suddenly change the current in a wire: Magnetic field will not change instantaneously. Electron and positron collide: Produce both electric and magnetic field, these fields exist even after annihilation. Changes propagate at speed of light There is no time in Biot-Savart law: Speed of moving charges must be small

  33. Electron Current A steady flow of charges in one direction will create a magnetic field. How can we cause charges to flow steadily? Need to find a way to produce and sustain E in a wire. Use battery

  34. Electron Current mobile electron density wire Cross sectional area Average drift speed Electron current:

  35. Typical Mobile Electron Drift Speed Typical electron current in a circuit is ~ 1018 electrons/s. What is the drift speed of an electron in a 1 mm thick copper wire of circular cross section?

  36. Typical Mobile Electron Drift Speed How much time would it take for a particular electron to move through a piece of wire 30 cm long? How can a lamp light up as soon as you turn it on?

  37. Conventional Current In some materials current moving charges are positive: Ionic solution “Holes” in some materials (same charge as electron but +) Observing magnetic field around copper wire: Can we tell whether the current consists of electrons or positive ‘holes’? The prediction of the Biot-Savart law is exactly the same in either case.

  38. Conventional Current André Marie Ampère (1775 - 1836) Metals: current consists of electrons Semiconductors: n-type – electrons p-type – positive holes Most effects are insensitive to the sign of mobile charges: introduceconventional current: Units: C/s  A (Ampere)

  39. Exercise A typical electron current in a circuit is 1018 electrons/s. What is the conventional current?

  40. The Biot-Savart Law for Currents Superposition principle is valid The Biot-Savart law for a short length of thin wire

  41. Biot-Savart Law Single Charge: The Biot-Savart law for a short length of thin wire Current: Moving charge produces a curly magnetic field B units: T (Tesla) = kg s-2A-1

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