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,

Magnetic Force The 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 Force The 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 B B v

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. + - v F F B B v

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 the before mentioned equation. 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

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

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. i i d xxxx xxxx xxxx xxxx B Dimensions: Cross sectional area: A Length: x

Hall Effect Electrons drift with a drift velocity v d as shown. When the magnetic field is turned on the electrons are deflected upwards. i i d xxxx xxxx xxxx xxxx B - vdvd FBFB FBFB

Hall Effect As time goes on electrons build up making on side –ve and the other +ve. i i d xxxx xxxx xxxx xxxx B - vdvd High Low FBFB

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. i i xxxx xxxx xxxx xxxx B - vdvd High Low FBFB E FEFE

Hall Effect The electric field pushed the electrons downwards. The continues until equilibrium where the electric force just cancels the magnetic force. i i xxxx xxxx xxxx xxxx B - vdvd High Low FBFB E FEFE

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. i i xxxx xxxx xxxx xxxx B - vdvd High Low FBFB E FEFE

Hall Effect The hall potential V is given by, V=Ed

Hall Effect When in balance,

Hall Effect When in balance, Recall, dx A A wire

Hall Effect Substituting for E, v d into we get,

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

Recall: 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 Force Consider a wire of length L, in a magnetic field, through which a current I passes. xx x x xxx x I B

Magnetic Force Consider 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, xx x x xxx x I B

Magnetic Force Thus we can write the force acting on the wire,

Magnetic Force Thus we can write the force acting on the wire, In general,

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.

Magnetic Force The 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 Force side1 side4 side2 side3 b a side1 side2 side3 rotation n b

Magnetic Force The 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 it’s perpendicular to the loops plane. side1 side2 side3 rotation n b θ

Magnetic Force The net force acting on the loop is the sum of the forces on each side. Clearly F 2 and F 4 cancel. However F 1 and F 3 act together to produce a torque. side1 side2 side3 rotation n b

The torque acts to rotate the loop so that n lines up with B. The torque to each is given by F x d. ie. The net torque, If there are N loops,

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:

Biot-Savart Law Where is the permeability of free space. And 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 Sol: l

Biot-Savart Law We rewrite the equation in terms of the angle the line extrapolated from makes with x-axis at the point P. Why? Because it’s more useful. l

Biot-Savart Law Sol: From the diagram, And hence l

Biot-Savart Law Sol: From the diagram, And hence l

Biot-Savart Law Hence, As well, Therefore, l

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

Biot-Savart Law For the case where B is due to a length AB, If AB is taken to infinity,