Chapter 26: Magnetism: Force and Field

Chapter 26: Magnetism: Force and Field • Magnets Magnetism

Magnetic forces Magnetism

Magnetic field of Earth Magnetism

S N S N S N • Magnetic monopoles? Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or magnetic charge). How can you isolate this magnetic charge? Try cutting a bar magnet in half: Magnetism Even an individual electron has a magnetic “dipole”! • Many searches for magnetic monopoles—the existence of which would explain (within framework of QM) the quantization of electric charge (argument of Dirac) • No monopoles have ever been found:

Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect) • Source of magnetic field • What is the source of magnetic fields, if not magnetic charge? • Answer: electric charge in motion! • e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. • Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Magnetism

Magnetic force: Observations Magnetism

Magnetic force (Lorentz force) Magnetism

Magnetic force (cont’d) Components of the magnetic force Magnetism

B B B x x x x x x x x x x x x x x x x x x ® ® ® ® ® ® ® ® ® ® v v v ´ q q q F F = 0 F • Magnetic force (cont’d) Magnetic force Magnetism

Magnetic force (cont’d) Units of magnetic field Magnetism

Magnetic force vs. electric force Magnetism

Magnetic Field Lines and Flux • Magnetic field lines

Magnetic field lines Magnetic Field Lines and Flux

Magnetic Field Lines and Flux • Magnetic field lines (cont’d)

Electric Field Linesof an Electric Dipole Magnetic Field Lines of a bar magnet Magnetic Field Lines and Flux • Magnetic field lines (cont’d)

Magnetic Field Lines and Flux • Magnetic field lines (cont’d)

B Area A  B B Magnetic Field Lines and Flux • Magnetic flux magnetic flux through a surface

Magnetic Field Lines and Flux • Magnetic flux (cont’d) Units: A=C/s, T=N/[C(m/s)] -> Tm2=Nm/[C/s]=Nm/A • Gauss’s law for magnetism No magnetic monopole has been observed!

Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field υ perpendicular to B The particle moves at constant speed υ in a circle in the plane perpendicular to B. F/m = a provides the acceleration to the center, so v R F B x

Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field

Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field Velocity selector

Motion of Charged Particles in a Magnetic Field • Case 1: Velocity perpendicular to magnetic field Mass spectrometer

Motion of Charged Particles in a Magnetic Field • Case 2: General case (vat any angle to B)

Motion of Charged Particles in a Magnetic Field • Case 2: General case (cont’d) Since the magnetic field does not exert force on a charge that travels in its direction, the component of velocity in the magnetic field direction does not change.

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current (straight wire)

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current (straight wire) (cont’d)

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current (curved wire)

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current: Example1

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current: Example1 (cont’d)

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current: Example2

Magnetic Force on a Current-Carrying Conductor • Magnetic force on a current: Example2 (cont’d)

Force and Torque on a Current Loop • Plane of loop is parallel to the magnetic field

Force and Torque on a Current Loop • Plane of loop : general case if q=90o

Force and Torque on a Current Loop • Plane of loop and magnetic moment

Force and Torque on a Current Loop • Plane of loop : magnetic moment (cont’d) The same magnetic dipole moment formulae work for any shape of planar loop. Any such loop can be filled by a rectangular mesh as in the sketch. Each sub-loop is made to carry the current NI. You will now see that all the interior wires have zero current and are of no consequence. Nevertheless, each sub-loop contributes to μ in proportion to its area.

Force and Torque on a Current Loop • Plane of loop : magnetic moment (cont’d)

Force and Torque on a Current Loop • Potential energy of a magnetic dipole Work done by the torque when the magnetic moment is rotated by df : In analogy to the case of an electric dipole in Chapter 22, we define a potential energy: Potential energy of a magnetic dipole at angle f to a magnetic field

Current increased • μ=I • Area increases • Torque fromBincreases • Angle of needle increases Current decreased • μdecreases • Torque fromBdecreases • Angle of needle decreases Applications • Galvanometer We have seen that a magnet can exert a torque on a loop of current – aligns the loop’s “dipole moment” with the field. • In this picture the loop (and hence the needle) wants to rotate clockwise • The spring produces a torque in the opposite direction • The needle will sit at its equilibrium position

Applications • Motor Slightly tip the loop Restoring force from the magnetic torque Oscillations Now turn the current off, just as the loop’s μ is aligned with B Loop “coasts” around until itsμ is ~antialigned withB Turn current back on Magnetic torque gives another kick to the loop Continuous rotation in steady state

VS I t Applications • Motor (cont’d) • Even better • Have the current change directions every half rotation • Torque acts the entire time • Two ways to change current in loop: • Use a fixed voltage, but change the circuit (e.g., break connection every half cycle •  DC motors • 2. Keep the current fixed, oscillate the source voltage • AC motors

Applications • Motor (cont’d) flip the current direction

Applications charges accumulate (in this case electrons) • Hall effect - - - + + + Measuring Hall voltage (Hall emf) In a steady state qEH =qvdB Charges move sideways until the Hall field EH grows to balance the force due to the magnetic field: n can be measured

Chapter 26: Magnetism: Force and Field