Lecture_12 - Magnetic Field A compass needle turns and points in a particular direction there is something which interacts with it Magnetic field(B

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Unformatted text preview: 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 Magnetic fields are produced by moving charges Current in a wire: convenient source of magnetic field The electron current (i) is the number of electrons per second that enter a section of a conductor. The Magnetic Effects of Currents Conclusions: 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. rsted How does the field around a wire look like? Hans Christian rsted (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 Bwire = 2 10-5 T tan 27 Bwire 1 10-5 T Biot-Savart Law for a Single Charge Electric field of a point charge: E = 1 q ^ r 2 4 0 r Moving charge makes a curly magnetic field: B units: T (tesla) = kg s-2A-1 0 qv r ^ B= 4 r 2 0 T m2 = 10-7 4 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 Calculate cross product (vector): Ax Bx Ay Bz - Az By A B = A y By = Az Bx - Ax Bz A B A B - A B z z x y y x Calculate magnitude: A B = A B sin Calculate direction: Right-hand rule 0 qv r ^ B= 4 r 2 Two-dimensional Projections a vector (arrow) is facing into the screen a vector (arrow) is facing out of the screen 0 qv r ^ B= 4 r 2 B B r v B B B Exercise 0 qv r ^ B= 4 r 2 What is B straight ahead? What if the charge is negative? Exercise What is the magnetic field created by an electron spinning around the nucleus in the simple Bohr model of the H atom? v = 2.2106 m/s r = 0.510-10 m v 0 qv r ^ B= 4 r 2 r 0 qv B= sin 2 4 r 2 1.6 10 -19 C 2.2 10 6 m/s -7 T m B = 10 sin 2 -10 C m/s 2 0.5 10 m ( ( )( ) ) B = 14 T (BEarth=210-5 T) Distance Dependence B2 0 qv r ^ B= 4 r 2 r B1 v B3 0 qv B= sin 2 4 r Which of the B1 and B2 is larger? Which of the B2 and B3 is larger? Moving Charge Sign Dependence 0 qv r ^ B= 4 r 2 r B1 v 0 qv B= sin 2 4 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. r B v B1 r v Exercise A current-carrying wire lies on top of a compass. What is the direction of the electron current in this wire? Find out experimentally from which end of the battery electrons emerge? What is the direction of conventional current? Frame of Reference Electric fields: produced by charges Magnetic fields: produced by moving charges Any magnetic field? 0 qv r ^ B= =0 4 r 2 0 qv r ^ B= 0 2 4 r charged tape Frame of Reference 0 qv r ^ B= 4 r 2 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 0 qv r ^ B= 4 r 2 There is no time in Biot-Savart law: Speed of moving charges must be small Electron Current To observe magnetic fields: need to produce steady flow of charges in one direction Need to find a way to produce and sustain E in a wire Use battery Electron Current m electrons 2 n A m v ( t s ) = nAv t electrons 3 s m ( ) mobile electron density wire Cross sectional area Average drift speed # electrons = nAv Electron current: i = s Typical Mobile Electron Density Example: copper wire (Cu) Molar mass = 64 g Density = 9 g/cm3 = 9.103 kg/m3 Each atom gives one electron 0.064 kg 1.07 10-25 kg Mass of one atom: mCu 6 1023 Number of atoms in 1 m3: 9 103 kg/m3 n 8.4 1028 m-3 1.07 10-25 kg 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 2 n 8.4 1028 m-3 -3 2 3.14 (1 10 m ) D A= = 8 10 -7 m 2 4 4 1018 s-1 1018 s-1 v= = 1.5 10 -5 m/s nA 8.4 10 28 m -3 8 10 -7 m 2 ( )( ) 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? n 8.4 1028 m-3 v = 1.5 10-5 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 t= = = 2 104 s 5.5 hours! v 1.5 10-5 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'? 0 qv r 0 ev r ^ ^ ^ 0 ( -e) ( - v ) r B= B= = 2 2 4 r 4 r 4 r2 The prediction of the Biot-Savart law is exactly the same in either case. Conventional Current 0 ev r ^ ^ 0 ( -e) ( - v ) r B= = 2 4 r 4 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) Andr Marie Ampre (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 ( )( ) ...
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This note was uploaded on 04/02/2008 for the course PHYS 272 taught by Professor K during the Winter '07 term at Purdue University-West Lafayette.

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