1Bweek10. - Physics 1B Magnetism Class Notes Walter...

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1 Physics 1B Class Notes© Walter Gekelman Week 10 Magnetism We have studies the force law for a charge in an electric field F = q E . If a magnetic field is present the generalized law, called the Lorentz equation is: (1) F = q E + v × B ( ) The magnetic field is in tesla (or weber/m 2 ) in MKS and in Gauss in cgs systems of units. 1 tesla = 10 4 Gauss. This is a velocity dependant law. The only other time we saw velocity occur in a force law is when frictional drag is present in the equation of motion of a falling object. In rectangular coordinates: (1a) or The magnetic field now gives us a “preferred direction” in space as form equation (1) there is no magnetic force on particles moving along the magnetic field. We can write the velocity as v = v + v . Now (2) F = q E + v + v ( ) × B ( ) = q E + v ( ) × B ( ) Let us assume that B = B 0 ˆ k ( ) and there is no electric field. Then (3) From the z component we get m d v z dt = 0 ; v z = v z 0 . The z component of the velocity is not changed and remains whatever it was when the magnetic field appears. The other components are: m d v x dt = v y B 0 ; m d v y dt = v x B 0
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2 Take the derivative of the equation for d v x dt and substitute the second equation to get Now there are two equations for the x,y velocity components: (4) If we define ω c = qB 0 m we get two equations that we have seen before. The solutions are: (5) v x = A cos c t ( ) + D sin c t ( ) A and B are constants that depend on the initial conditions. Suppose at t=0 v = v x . This means that D=0. v x = A cos
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1Bweek10. - Physics 1B Magnetism Class Notes Walter...

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