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Transcript
Potential at a Certain Location 1. Add up the contribution of all point charges at this point 1 qi VA = å i 4pe 0 ri q1 q2 r2 r1 A 2. Travel along a path from point very far away to the location of interest and add up -E · dl at each step: q2 A dl E VA = - ò E · dl ¥ q1 A Common Pitfall Assume that the potential V at a location is defined by the electric field E at this location. A Example: E = 0 inside a charged metal sphere, but V is not! 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 at rest before 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 0.5m B A. B. C. D. E. 1.5J 0J –1.5J 3.0J –3.0J E = 5 N/C 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 rest before 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? A E = 5 N/C 0.5m A. B. C. D. E. 1.5J 0J –1.5J 3.0J -3.0J B C 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 rest before and after the move. How much work was done by the net external force while moving the test charge this time? A D 0.5m B A. B. C. D. E. 0.5m C 1.5J 0J –1.5J Infinitely big Do not know at this time. E = 5 N/C +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 rest before and after the move. What is the voltage difference between B and C, and at which point is the voltage larger? A D 0.5m B A. B. C. D. E. 0.5m 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. E = 5 N/C Chapter 18 Magnetic Field 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 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 Detecting Magnetic Fields We 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 objects polarize and interact with these materials Points toward North pole – electric dipole does not do that The Magnetic Effects of Currents Imagine an 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? The Magnetic Effects of Currents 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? Hans Christian Ørsted (1777 - 1851) The Magnetic Effects of Currents The moving electrons in a wire create a magnetic field Principle of superposition: Bnet = BEarth + Bwire What can you say about the magnitudes of BEarth and Bwire? What if BEarth were much larger than Bwire? 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). Bnet = BEarth + Bwire Bwire = BEarth tan q Bwire = 2 ´ 10-5 T ´ tan 27 Bwire » 1 ´ 10-5 T Biot-Savart Law for a Single Charge q Electric field of a point charge: E = rˆ 2 4pe 0 r 1 Moving charge makes a curly magnetic field: B units: T (tesla) = kg s-2A-1 m0 qv ´ rˆ B= 4p r 2 2 m0 -7 T × m = 10 4p C × m/s Jean-Baptiste Biot (1774-1862) Felix Savart (1791-1841) Nikola Tesla (1856-1943) Nikola Tesla (1856-1943) High tension coil demonstration The Cross Product æ A ö æ B ö æ A B -A B z y ç x ÷ ç x ÷ ç y z A ´ B = ç A y ÷ ´ ç By ÷ = ç Az Bx - Ax Bz çç ÷÷ çç ÷÷ ç ç è Az ø è Bz ø è Ax By - Ay Bx = ( A B - A B ) iˆ - ( A B - A B ) ĵ + ( A B y z z y x z z x x y ö ˆ i ÷ ÷ = Ax ÷ ÷ Bx ø - Ay Bx ) k̂ Calculate magnitude: A ´ B = A B sinq Calculate direction: Right-hand rule m0 qv ´ rˆ B= 4p r 2 ĵ k̂ Ay Az By Bz Question A´ B A =< 0,0, 3 >; B =< 0, 4,0 > What is the direction of < 0, 0, 3> x < 0, 4, 0>? 𝑖 𝑗 0 0 0 4 𝑘 3 = (0 − 12)𝑖 0 A) +x B) –x C) +y D) –y E) zero magnitude Two-dimensional Projections a vector (arrow) is facing into the screen a vector (arrow) is facing out of the screen m0 qv ´ rˆ B= 4p r 2 B B B r v B B Why must the field change direction above and below the dashed line? Exercise m0 qv ´ rˆ B= 4p r 2 What is B straight ahead? What if the charge is negative? Distance Dependence B2 m0 qv ´ rˆ B= 4p r 2 B1 B3 r v m0 qv B= sinq 2 4p r Which is larger, B1 or B3 ? Which is larger, B1 or B2 ? Moving Charge Sign Dependence m0 qv ´ rˆ B= 4p r 2 B1 r m0 qv B= sinq 2 4p r 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. v + B r - v B1 r v - 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 A) B) C) D) E) -x +x -z +z No magnetic field x v z 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? 𝐵 𝑒− 𝑒− 𝑒− 𝑒− Frame of Reference Electric fields: produced by charges Magnetic fields: produced by moving charges B= m0 qv ´ rˆ =0 2 4p r Any magnetic field? m0 qv ´ rˆ B= ¹0 2 4p r charged tape Frame of Reference m0 qv ´ rˆ B= 2 4p r 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) 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 m0 qv ´ rˆ B= 4p r 2 There is no time in Biot-Savart law: Speed of moving charges must be small 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 Electron Current mö æ electrons ö 2 æ çè n ÷ø A m çè v ÷ø ( Dt s ) = nAv Dt electrons 3 m s ( mobile electron density ) wire Cross sectional area Average drift speed # electrons Electron current: i = = nAv s 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? # electrons = nAv s n » 8.4 ´ 1028 m-3 3.14 × (1 ´ 10 m ) pD A= » = 8 ´ 10 -7 m 2 4 4 -3 2 2 1018 s-1 1018 s-1 -5 v= » = 1.5 ´ 10 m/s 28 -3 -7 2 nA 8.4 ´ 10 m 8 ´ 10 m ( )( ) Typical Mobile Electron Drift Speed -5 v = 1.5 ´ 10 m/s How much time would it take for a particular electron to move through a piece of wire 30 cm long? s 0.3 m 4 t= = = 2 ´ 10 s » 5.5 hours! -5 v 1.5 ´ 10 m/s How can a lamp light up as soon as you turn it on? 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’? m0 qv ´ rˆ B= 4p r 2 m0 ev ´ rˆ m0 ( -e) ( -v ) ´ r̂ B= = 2 4p r 4p r2 The prediction of the Biot-Savart law is exactly the same in either case. Conventional Current m0 ev ´ rˆ m0 ( -e) ( -v ) ´ r̂ B= = 2 4p r 4p r2 Metals: current consists of electrons Semiconductors: n-type – electrons p-type – positive holes Most effects are insensitive to the sign of mobile charges: introduce conventional current: I = q i = q nAv Units: C/s A (Ampere) André Marie Ampère (1775 - 1836) Exercise A typical electron current in a circuit is 1018 electrons/s. What is the conventional current? ( )( ) I = q i = 1.6 ´ 10 -19 C 1018 s-1 = 0.16 A The Biot-Savart Law for Currents Superposition principle is valid m0 qi vi ´ r̂i DBi = 4p ri2 I = q i = q nAv The Biot-Savart law for a short length of thin wire m0 qi vi ´ r̂i DB = å DBi = å 2 4 p r i i i m0 qv ´ rˆ DB = 1 å 2 4p r i m0 qv ´ rˆ DB = nADl 2 4p r m0 IDl ´ rˆ DB = 4p r 2 Biot-Savart Law Moving charge produces a curly magnetic field m0 qv ´ rˆ B= Single Charge: 4p r 2 Current: m0 IDl ´ rˆ The Biot-Savart law for a DB = short length of thin wire 4p r 2 I = q i = q nAv B units: T (Tesla) = kg s-2A-1 2 m0 T × m = 10-7 4p C × m/s