MagnetismMagnetic Force

Magnetic Force OutlineLorentz ForceCharged particles in a crossed fieldHall EffectCirculating charged particlesMotorsBio-Savart Law

Class ObjectivesDefine 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 ForceThe magnetic field is defined from the Lorentz Force Law,

Magnetic ForceThe magnetic field is defined from the Lorentz Force Law,Specifically, for a particle with charge q moving through a field B with a velocity v,

That is q times the cross product of v and B.

Magnetic ForceThe cross product may be rewritten so that,

The angle is measured from the direction of the velocity to the magnetic field .NB: the smallest angle between the vectors!v x BBv

Magnetic Force

Magnetic ForceThe diagrams show the direction of the force acting on a positive charge.The force acting on a negative charge is in the opposite direction.+-vFFBBv

Magnetic ForceThe 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 ForceThe SI unit for B is the tesla (T) newton per coulomb-meter per second and follows from the before mentioned equation .1 tesla = 1 N/(Cm/s)

Magnetic Field LinesReview

Magnetic Field LinesMagnetic 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 LinesCrossed Fields

Crossed FieldsBoth 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 FieldsExamples of crossed fields are: cathode ray tube, velocity selector, mass spectrometer.

Crossed FieldsHall Effect

Hall EffectAn interesting property of a conductor in a crossed field is the Hall effect.

Hall EffectAn 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.iidxxxxxxxxxxxxxxxxDimensions:

Cross sectional area: ALength: x

Hall EffectElectrons drift with a drift velocity vd as shown.When the magnetic field is turned on the electrons are deflected upwards.iidxxxxxxxxxxxxxxxx-vdFB

Hall EffectAs time goes on electrons build up making on side ve and the other +ve.iidxxxxxxxxxxxxxxxx-vd- - - - -+ + + + +High Low

Hall EffectAs time goes on electrons build up making on side ve and the other +ve.This creates an electric field from +ve to ve. iixxxxxxxxxxxxxxxx-vd- - - - -+ + + + +High Low

Hall EffectThe electric field pushed the electrons downwards.The continues until equilibrium where the electric force just cancels the magnetic force.iixxxxxxxxxxxxxxxx-vd- - - - -+ + + + +High Low

Hall EffectAt this point the electrons move along the conductor with no further collection at the top of the conductor and increase in E.

iixxxxxxxxxxxxxxxx-vd- - - - -+ + + + +High Low

Hall EffectThe hall potential V is given by, V=Ed

Hall EffectWhen in balance,

Hall EffectWhen in balance,

Recall,dxAA wire

Hall EffectSubstituting for E, vd into we get,

A circulating charged particle

Magnetic ForceA 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 ForceRecall: for a charged particle moving in a circle of radius R,

As so we can show that,

Magnetic Force on a current carrying wire

Magnetic ForceConsider a wire of length L, in a magnetic field, through which a current I passes.

Magnetic ForceConsider a wire of length L, in a magnetic field, through which a current I passes.

The force acting on an element of the wire dl is given by,

Magnetic ForceThus we can write the force acting on the wire,

Magnetic ForceThus we can write the force acting on the wire,

In general,

Magnetic ForceThe force on a wire can be extended to that on a current loop.

Magnetic ForceThe force on a wire can be extended to that on a current loop.An example of which is a motor.

Magnetic ForceThe force on a wire can be extended to that on a current loop.An example of which is a motor.The diagram on the next slide shows a simple motor made up of a rectangular loop of sides a and b carrying a current I.

Magnetic Forceside1side4side2side3ba

Magnetic ForceThe loop is oriented so that S1 and S3 perpendicular to the magnetic field and S2 and S4 are not.The vector n is defined so that its perpendicular to the loops plane.

Magnetic ForceThe net force acting on the loop is the sum of the forces on each side.Clearly F2 and F4 cancel.However F1 and F3 act together to produce a torque.

The torque acts to rotate the loop so that n lines up with B.The torque to each is given by Fx d. ie.

The net torque,

If there are N loops,

InterludeNext.The Biot-Savart Law

Biot-Savart Law

ObjectiveInvestigate the magnetic field due to a current carrying conductor.Define the Biot-Savart LawUse the law of Biot-Savart to find the magnetic field due to a wire.

Biot-Savart LawSo 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 LawWe will illustrate the Biot-Savart Law.

Biot-Savart LawBiot-Savart law:

Biot-Savart LawWhere is the permeability of free space.

And is the vector from dl to the point P.

Biot-Savart LawExample: Find B at a point P from a long straight wire.

Biot-Savart LawSol:

Biot-Savart LawWe rewrite the equation in terms of the angle the line extrapolated from makes with x-axis at the point P.Why?Because its more useful.

Biot-Savart LawSol:From the diagram,

And hence

Biot-Savart LawSol:From the diagram,

And hence

Biot-Savart LawHence,As well,

Therefore,

Biot-Savart LawFor the case where B is due to a length AB,

Biot-Savart LawFor the case where B is due to a length AB,

If AB is taken to infinity,

*Beta -> +90 and alpha ->-90