Section Summary

Section Summary

5.1 Magnets

  • Magnetism is a subject that includes the properties of magnets, the effect of the magnetic force on moving charges and currents, and the creation of magnetic fields by currents.
  • There are two types of magnetic poles, called the north magnetic pole and south magnetic pole.
  • North magnetic poles are those that are attracted toward Earth’s geographic North Pole.
  • Like poles repel and unlike poles attract.
  • Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.

5.2 Magnetic Fields and Magnetic Field Lines

  • Magnetic fields can be pictorially represented by magnetic field lines, the properties of which are as follows:
  1. The field is tangent to the magnetic field line.
  2. Field strength is proportional to the line density.
  3. Field lines cannot cross.
  4. Field lines are continuous loops.

5.3 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field

  • Magnetic fields exert a force on a moving charge q, the magnitude of which is
    F=qvBsinθ,F=qvBsinθ, size 12{F= ital "qvB""sin"θ} {}
    where θθ size 12{θ} {} is the angle between the directions of vv size 12{v} {} and B.B.size 12{B} {}
  • The SI unit for magnetic field strength BB size 12{B} {} is the tesla (T), which is related to other units by
    1 T=1 NCm/s=1 NAm.1 T=1 NCm/s=1 NAm.
  • The direction of the force on a moving charge is given by right-hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of v,v,size 12{v} {} the fingers in the direction of B,B,size 12{B} {} and a perpendicular to the palm points in the direction of F.F.size 12{F} {}
  • The force is perpendicular to the plane formed by vv and B.B.size 12{B} {} Because the force is zero if vv size 12{v} {} is parallel to B,B,size 12{B} {} charged particles often follow magnetic field lines rather than cross them.

5.4 Force on a Moving Charge in a Magnetic Field: Examples and Applications

  • Magnetic force can supply centripetal force and cause a charged particle to move in a circular path of radius
    r=mvqB,r=mvqB, size 12{r= { { ital "mv"} over { ital "qB"} } ,} {}
    where vv size 12{v} {} is the component of the velocity perpendicular to BB size 12{B} {} for a charged particle with mass mm size 12{m} {} and charge qq size 12{q} {}.

5.5 The Hall Effect

  • The Hall effect is the creation of voltage ε,ε,size 12{ε} {} known as the Hall emf, across a current-carrying conductor by a magnetic field.
  • The Hall emf is given by
    ε=Blv(B,v,andl,mutually perpendicular)ε=Blv(B,v,andl,mutually perpendicular) size 12{ε= ital "Blv"``` \( B,`v,`"and"`l,`"mutually perpendicular" \) } {}
    for a conductor of width ll size 12{l} {} through which charges move at a speed v.v.size 12{v} {}

5.6 Magnetic Force on a Current-Carrying Conductor

  • The magnetic force on current-carrying conductors is given by
    F=IlBsinθ,F=IlBsinθ, size 12{F= ital "IlB""sin"θ} {}
    where II size 12{I} {} is the current, ll size 12{l} {} is the length of a straight conductor in a uniform magnetic field B,B, size 12{B} {} and θθ size 12{θ} {} is the angle between II size 12{I} {} and B.B. size 12{B} {} The force follows RHR-1 with the thumb in the direction of I.I. size 12{I} {}

5.7 Torque on a Current Loop: Motors and Meters

  • The torque ττ size 12{τ} {} on a current-carrying loop of any shape in a uniform magnetic field. is
    τ=NIABsinθ,τ=NIABsinθ, size 12{τ= ital "NIAB""sin"θ} {}
    where NN size 12{N} {} is the number of turns, II size 12{I} {} is the current, AA size 12{A} {} is the area of the loop, BB size 12{B} {} is the magnetic field strength, and θθ size 12{θ} {} is the angle between the perpendicular to the loop and the magnetic field.

5.8 Magnetic Fields Produced by Currents: Ampere's Law

  • The strength of the magnetic field created by current in a long straight wire is given by
    B=μ0I2πr(long straight wire),B=μ0I2πr(long straight wire),
    II size 12{I} {} is the current, rr size 12{r} {} is the shortest distance to the wire, and the constant μ0=×107Tm/Aμ0=×107Tm/A size 12{μ rSub { size 8{0} } =4π times "10" rSup { size 8{ - 7} } `T cdot "m/A"} {} is the permeability of free space.
  • The direction of the magnetic field created by a long straight wire is given by right-hand rule 2 (RHR-2): Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.
  • The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampere’s law.
  • The magnetic field strength at the center of a circular loop is given by
    B=μ0I2R( at center of loop),B=μ0I2R( at center of loop), size 12{B= { {μ rSub { size 8{0} } I} over {2R} } " " \( "at center of loop" \) ,} {}
    RR size 12{R} {} is the radius of the loop. This equation becomes B=μ0nI/(2R)B=μ0nI/(2R) size 12{B=μ rSub { size 8{0} } ital "nI"/ \( 2R \) } {} for a flat coil of NN size 12{N} {} loops. RHR-2 gives the direction of the field about the loop. A long coil is called a solenoid.
  • The magnetic field strength inside a solenoid is
    B=μ0nI(inside a solenoid),B=μ0nI(inside a solenoid), size 12{B=μ rSub { size 8{0} } ital "nI"" " \( "inside a solenoid" \) ,} {}
    where nn size 12{n} {} is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.

5.9 Magnetic Force between Two Parallel Conductors

  • The force between two parallel currents I1I1 size 12{I rSub { size 8{1} } } {} and I2,I2,size 12{I rSub { size 8{2} } } {} separated by a distance r,r,size 12{r} {} has a magnitude per unit length given by
    Fl=μ0I1I22πr.Fl=μ0I1I22πr. size 12{ { {F} over {l} } = { {μ rSub { size 8{0} } I rSub { size 8{1} } I rSub { size 8{2} } } over {2πr} } } {}
  • The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.

5.10 More Applications of Magnetism

  • Crossed (perpendicular) electric and magnetic fields act as a velocity filter, giving equal and opposite forces on any charge with velocity perpendicular to the fields and of magnitude
    v=EB.v=EB. size 12{v= { {E} over {B} } "." } {}