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Magnetism Magnetic Force Magnetic Force Outline • • • • • • Lorentz Force Charged particles in a crossed field Hall Effect Circulating charged particles Motors Bio-Savart Law Class Objectives • Define the Lorentz Force equation. • Show it can be used to find the magnitude and direction of the force. • Quickly review field lines. • Define cross fields. • Hall effect produced by a crossed field. • Derive the equation for the Hall voltage. Magnetic Force • The magnetic field is defined from the Lorentz Force Law,F qE q v B Magnetic Force • The magnetic field is defined from the Lorentz Force Law,F qE q v B • Specifically, for a particle with charge q moving through a field B with a velocity v, F q vB • That is q times the cross product of v and B. Magnetic Force • The cross product may be rewritten so that, F q vBsin • The angle is measured from the direction of the velocity v to the magnetic field B . • NB: the smallest angle between the vectors! vxB B v Magnetic Force Magnetic Force • The diagrams show the direction of the force acting on a positive charge. • The force acting on a negative charge is in the opposite direction. B F - v B + v F Magnetic Force • The direction of the force F acting on a charged particle moving with velocity v through a magnetic field B is always perpendicular to v and B. Magnetic Force • The SI unit for B is the tesla (T) newton per coulomb-meter per second and follows from F the before mentioned equation q v sin B. • 1 tesla = 1 N/(Cm/s) Magnetic Field Lines Review Magnetic Field Lines • Magnetic field lines are used to represent the magnetic field, similar to electric field lines to represent the electric field. • The magnetic field for various magnets are shown on the next slide. Magnetic Field Lines Crossed Fields Crossed Fields • Both an electric field E and a magnetic field B can act on a charged particle. When they act perpendicular to each other they are said to be ‘crossed fields’. Crossed Fields • Examples of crossed fields are: cathode ray tube, velocity selector, mass spectrometer. Crossed Fields Hall Effect Hall Effect • An interesting property of a conductor in a crossed field is the Hall effect. Hall Effect • An interesting property of a conductor in a crossed field is the Hall effect. • Consider a conductor of width d carrying a current i in a magnetic field B as shown. d i x x x x x x x x x x x x x Bx x x Dimensions: i Cross sectional area: A Length: x Hall Effect • Electrons drift with a drift velocity vd as shown. • When the magnetic field is turned on the electrons are deflected upwards. d i FB x x x x x x FB x x x x x x x Bx x x vd - i Hall Effect • As time goes on electrons build up making on side –ve and the other +ve. x d i - x +x x - x x x FB x +x + x vd Bx x x - x Low - + x + x i High Hall Effect • As time goes on electrons build up making on side –ve and the other +ve. • This creates an electric field from +ve to –ve. x i E x +x x - x x x FB x + x FE+ x vd Bx x x - x Low - + x + x i High Hall Effect • The electric field pushed the electrons downwards. • The continues until equilibrium where the electric force just cancels the magnetic x x x Low force. x - i E x +x x - x FB x + x FE+ x vd Bx x - x - + x + x i High Hall Effect • At this point the electrons move along the conductor with no further collection at the top of the conductor and increase in E. x i E x +x x - x x x FB x + x FE+ x vd Bx x x - x Low - + x + x i High Hall Effect • The hall potential V is given by, V=Ed Hall Effect • When in balance, FB FE eE evd B i where vd neA Hall Effect • When in balance, FB FE eE evd B i where vd neA dq • Recall, i dt dq neAdx dx neAvd i neA dt dx A A wire Hall Effect • Substituting for E, vd into eE evd B we get, Bi n Vle A where l d A circulating charged particle Magnetic Force • A charged particle moving in a plane perpendicular to a magnetic field will move in a circular orbit. • The magnetic force acts as a centripetal force. • Its direction is given by the right hand rule. Magnetic Force Magnetic Force • Recall: for a charged particle moving in a 2 circle of radius R,FB mv mv qvB R mv R qB 2 R 2m • As so we can show that,T qB qB qB ,f , 2m m Magnetic Force on a current carrying wire Magnetic Force • Consider a wire of length L, in a magnetic field, through which a current I passes. x x x x B x x x x I Magnetic Force • Consider a wire of length L, in a magnetic field, through which a current I passes. x x x x B x x x x I • The force actingon an element of the wire dl is given by,dFB IdL B Magnetic Force • Thus we can write the force acting on the wire, dFB BIdL L FB BI dL 0 FB BIL Magnetic Force • Thus we can write the force acting on the wire, dFB BIdL L FB BI dL 0 FB BIL • In general, FB BIL sin Magnetic Force • The force on a wire can be extended to that on a current loop. Magnetic Force • The force on a wire can be extended to that on a current loop. • An example of which is a motor. Interlude Next…. The Biot-Savart Law Biot-Savart Law Objective • Investigate the magnetic field due to a current carrying conductor. • Define the Biot-Savart Law • Use the law of Biot-Savart to find the magnetic field due to a wire. Biot-Savart Law • So far we have only considered a wire in an external field B. Using Biot-Savart law we find the field at a point due to the wire. Biot-Savart Law • We will illustrate the Biot-Savart Law. Biot-Savart Law • Biot-Savart law: 0 I ˆ dB d l r 4r 2 0 Idl sin dB 2 4r Biot-Savart Law • Where 0 is the permeability of free space. 0 4 10 7 Tm / A • And r̂ is the vector from dl to the point P. Biot-Savart Law • Example: Find B at a point P from a long straight wire. l Biot-Savart Law 0 I Idl sin 0 dl rˆ dB • Sol: dB 2 4r 4r 2 l Biot-Savart Law • We rewrite the equation in terms of the angle the line extrapolated from r̂ makes with x-axis at the point P. • Why? • Because it’s more useful. l Biot-Savart Law 0 I Idl sin 0 dl rˆ dB • Sol: dB 2 4r 4r 2 • From the diagram, 180 • And hence 90 l Biot-Savart Law 0 I Idl sin 0 dl rˆ dB • Sol: dB 2 4r 4r 2 • From the diagram, 180 • And hence 90 sin sin 90 cos sinA B sin A cos B sin B cos A l Biot-Savart Law 0 Idl cos dB 4r 2 • Hence, • As well, l tan x x cos r l r x2 l 2 0 I cos d • Therefore, dB 4x Biot-Savart Law • For the case where B is due to a length AB, B 0i B dB cos d 4x 0 0i sin sin 4x A B Biot-Savart Law • For the case where B is due to a length AB, B 0i B dB cos d 4x 0 0i sin sin 4x • If AB is taken to infinity, 0i B 2x A B