Magnetism

• Electromagnetism


  Part of CCP1200
• Magnetic fields


  Fields that exist around permanent magnets, moving charges and wires carrying electric currents

  
• Magnetic fields do not start on monopoles (positive and negative charges for electric fields), but are rather in the shape of loops

  
• Classically, magnetic monopoles do not exist
• Magnetic force on a charge q moving in a magnetic field ⃗B with velocity ⃗v


  ⃗FB = q⃗v × ⃗B

  
• Magnetic field magnitude

  Measured in SI units of 1 Tesla = 1 Ns/Cm=1 N/Am, but non-SI units of 1 Gauss = 1 G = 10−4 T are also often used

  
• Lorentz force


  The force that a charge will experience in regions of space where both electric and magnetic fields are present: ⃗F = q ⃗E + q⃗v × ⃗B

  
• The magnitude of the magnetic force is given by F = |q|vB sin ϕ, where ϕ is the angle between the direction of charge travel and the direction of the magnetic field

  
• The magnetic force is always perpendicular to the direction of charge motion

  
• Magnetic forces are always centripetal, they cause no change in the magnitude of the velocity of the charge, and therefore do no work
• Motion of a charge in a region of magnetic field

  Question: A charge of mass m and charge q is observed travelling through a region of uniform magnetic field with velocity ⃗v = vxxˆ + vyyˆ + vzzˆ. The magnetic field has magnitude B and is oriented along the positive z direction. Describe the motion of the particle using the quantities given.

  1. The z-component of the velocity remains unaffected, causing the particle to translate along the z direction with constant velocity

  2. The perpendicular component causes continuous circular motion of the charge in the xy plane, with radius R = mv⊥/qB

  
• Thomson's experiment


  Used the Lorentz force to obtain a value for electron charge-to-mass ratio

  
• Thomson concluded that the charges were negative based on the downwards deflection of the electron beam in the magnetic field
• Determining the electron charge-to-mass ratio in Thomson's experiment


  1. Use energy conservation to relate the electron velocity to the applied voltage

  2. Set the net force on the electrons to zero when both electric and magnetic fields are applied, to obtain the e/m ratio

  
• The high value of e/m ratio obtained by Thomson led to the conclusion that the mass of the electrons must be extremely small
• Magnetic force on a current-carrying wire


  ⃗FB = integral of wire d F ⃗B

  = integral of wire I dL × ⃗B

  
• Uniform magnetic fields do not cause any net forces on loops of current-carrying wires
• Magnetic moment of a current loop

  ⃗µ = I ⃗A

  
• Net torque on a current loop in a magnetic field

  ⃗τ = ⃗µ × ⃗B

  - Torque = the force that causes an object to rotate about an axis

  
• A current loop in a magnetic field will experience a torque that will rotate it until its magnetic moment aligns with the magnetic field
• Net force on a circular current loop in a uniform magnetic field


  The net force is zero, as the magnetic force on each segment of the wire cancels out
• Net torque on a square current loop in a uniform magnetic field

  The torque is ⃗τ = −abIBˆz, which will rotate the loop
• Magnetic field generated by a moving charge

  ⃗B = µ0/4π x (q⃗v × ˆr)r^2

  -A discrete charge q moving at a constant velocity ⃗v in free space will generate magnetic field B⃗

  
• Magnetic field generated by a current-carrying wire: Biot and Savart law

  ⃗B = µ0/4π x integral of wire (Id⃗L × ˆr)/r^2

  
• Adjacent current-carrying wires will experience attractive or repulsive forces between them, depending on the relative direction of the currents
• Loops of currents will experience magnetic forces, both individually (different sides of the loop acting to stretch it) and between them (different loops of currents attracting or repelling each other)

  
• Atoms are composed of positive nuclei and a cloud of electrons, and spin is a fundamental property of the electrons
• ˆr

  Unit vector connecting the wire segment under consideration to the observation point
• Magnetic forces between current carrying wires


  • Current carrying wires generate magnetic fields
  • Current carrying wires experience magnetic forces in the presence of external magnetic fields
  • Adjacent wires experience attractive or repulsive forces depending on the relative flow of currents

  
• Loops of currents

  • Experience magnetic forces individually (different sides of the loop acting to stretch it)
  • Experience magnetic forces between them (different loops of currents attracting or repelling each other)

  
• Magnetic moments exert forces on each other
• Magnetic materials

  • Atoms have electrons with spin, which gives them an intrinsic magnetic moment
  • Atoms can also have non-zero orbital angular momentum
  • Materials with atoms that have non-zero magnetic moments can form domains where the magnetic moments are aligned
  • If the overall alignment of the domains results in a non-zero magnetic moment, the material is magnetic and generates a magnetic field

  
• Ferromagnetism


  Permanent magnetisation of a material due to strong interactions between magnetic domains
• Paramagnetism and diamagnetism

  Magnetisation is lost upon removal of external magnetic field due to weaker dipole interactions compared to randomising thermal influences

  • Para = weakly attractive
  • Dia = repelled
• Calculating magnetic field due to a straight current-carrying segment


  1. Consider a small segment dx

  2. Use Biot-Savart law to get contribution dB

  3. Integrate along the length of the wire to get total B

  
• Calculating force between parallel current-carrying wires


  1. Each wire generates a magnetic field

  2. Use force on a current-carrying wire in a magnetic field to get force per unit length
• Calculating magnetic field due to a circular loop of current


  1. Consider a small segment dL of the loop

  2. Use Biot-Savart law to get contribution dB⊥

  3. Integrate around the loop to get total B

  
• Right Hand Rules (RHR)

  • For determining directions of vectors in cross-product expressions
  • For applying properties of cross-products in geometries with radial symmetry
• Magnetic field lines


  Visualize magnetic fields, density and direction determine magnitude and direction of magnetic field

  
• Magnetic field lines are always observed as loops, unlike electric field lines