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force_motional

force_motional - EE201.3 Class Notes Force on a moving...

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EE201.3 Class Notes Denard Lynch Page 1 of 4 September 2007 Force on a moving charge in a magnetic field (Note: see the last page (p4) of this section for a refresher on cross-products if required.) Recall the two sides of the relationship between electricity and magnetism: i) a moving charge creates a magnetic field, and ii) a magnetic field exerts a force on a a moving charge We are now going to explore part ii) of this relationship. First, highlight the differences between electric and magnetic fields w.r.t. their effect on a charged particle: Electric Fields Magnetic Fields F e is in the direction of the field F m is perpendicular to the field F e will act on a particle regardless of velocity, and will accelerate a particle F m only affects a charge in motion, but will not change its velocity (i.e. no acceleration) F e does work to displace/move a charge, and the energy of the charge is affected F m does no work on the charge, and the particles energy is not affected Recall: Work = Force acting through a distance. F d S = F v dt = 0, i.e. no work is done since there is no force in the direction of the motion. To explore F m a little further, from physics we know: F m ! q , vel , B , where: B = strength of the magnetic field, vel is the velocity of the particle(s), and q is the magnitude of the charge. Also, F m is perpendicular to both vel and B . F m = q ! vel " B (1) Now consider a collection of charged particles in a medium in the presence of a magnetic field of concentration B. If the charges and the magnetic field are static , there is no force or effect on the charged particles. However, the charged particles are all ‘drifting’ with a velocity of v drift , then the force on each one is given by (1) above, where q is the magnitude of the charge on one particle, vel is its velocity and B is the density of the field. The total number of charge carrying particles in the medium (e.g., a wire/conductor) is equal to nAl , where n = number of particles/m 3 , A = cross-sectional area in m 2 , and l is the length of the material in m. Then the total force (due to B) on the medium (wire) is: F m ! total = q " vel # B ( ) nAl ( ) (2) where q is the magnitude of charge per carrier in coulombs/carrier n is the density of the charge carriers per m 3 A is the cross-sectional area in m 2 , and v drift is the velocity of the charges in m/s. and: I = qnAv drift (3) F m B vel

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EE201.3 Class Notes Denard Lynch Page 2 of 4 September 2007 (Note: that all units in the expression cancel to leave coulombs/sec, and 1 c/s = 1 ampere.) Substituting (3) into (2) results in the expression for the force on a wire in a field, B: F mag = Il ! B (4) where I is the current in A, l is the length of the wire in the filed in meters, m, and B is the magnetic field density in Teslas, T.
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