chapter 4 - Moving Charges And Magnetism

• Both Electricity and Magnetism have been known for more than 2000 years
• It was realised that electricity and magnetism were intimately related

  1820
• Oersted's discovery

  • A current in a straight wire caused a noticeable deflection in a nearby magnetic compass needle
  • The alignment of the needle is tangential to an imaginary circle which has the straight wire as its centre and has its plane perpendicular to the wire
  • Reversing the direction of the current reverses the orientation of the needle
  • The deflection increases on increasing the current or bringing the needle closer to the wire
  • Iron filings sprinkled around the wire arrange themselves in concentric circles with the wire as the centre
• Oersted concluded that moving charges or currents produced a magnetic field in the surrounding space
• The laws obeyed by electricity and magnetism were unified and formulated by James Maxwell

  1864
• James Maxwell realised that light was electromagnetic waves
• Radio waves were discovered by Hertz, and produced by J.C.Bose and G. Marconi by the end of the 19th century
• A remarkable scientific and technological progress took place in the 20th century due to our increased understanding of electromagnetism and the invention of devices for production, amplification, transmission and detection of electromagnetic waves
• Magnetic field B

  A vector field produced by moving charges or currents
• Lorentz force

  The force on a charged particle due to both electric and magnetic fields, given by F = q(E + v x B)
• Magnetic force on a moving charge

  • Depends on q, v and B
  • The force is perpendicular to both v and B
  • The force is zero if v and B are parallel or anti-parallel
• Tesla (T)

  The unit of magnetic field, when the force acting on a unit charge (1 C), moving perpendicular to B with a speed 1m/s, is one newton
• Gauss

  A smaller non-SI unit of magnetic field, 1 gauss = 10^-4 tesla
• The earth's magnetic field is about 3.6 × 10^-5 T
• Magnetic force on a current-carrying conductor

  1. F = IlxB

  2. Where I is the current, l is the length vector of the conductor, and B is the external magnetic field
• Magnetic force on a straight wire

  • A straight wire of mass 200 g and length 1.5 m carries a current of 2 A. It is suspended in mid-air by a uniform horizontal magnetic field B. The magnitude of the magnetic field is 0.65 T
• Charged particle moving in a magnetic field

  For an electron, the Lorentz force is along the -z axis

  For a proton, the Lorentz force is along the +z axis
• Motion of a charged particle in a magnetic field

  1. Velocity v perpendicular to B: Produces circular motion perpendicular to magnetic field

  2. Velocity has component along B: Produces helical motion
• Centripetal force

  Force m v^2/r that acts perpendicular to the path towards the centre of the circle
• The direction of the Lorentz force on an electron (negative charge) moving along the positive x-axis in a magnetic field along the y-axis is along the -z axis
• The direction of the Lorentz force on a proton (positive charge) moving along the positive x-axis in a magnetic field along the y-axis is along the +z axis
• Biot-Savart law: The magnetic field dB produced by a current element I dl at a distance r is proportional to I dl and inversely proportional to r^2
• The magnetic field on the y-axis at a distance of 0.5 m from a current element of 10 A and length 1 cm placed at the origin is 4 × 10^-8 T in the +z direction
• The magnetic field at the centre of a circular current loop is B = (μ_0 I) / (2 R)
• The direction of the magnetic field due to a current loop is given by the right-hand thumb rule
• Magnetic field at P due to entire circular loop

  2πR * μ0 * I / (2(x^2 + R^2)^(3/2))
• Magnetic field at centre of loop


  μ0 * I / (2R)
• Magnetic field lines due to a circular wire form closed loops
• Right-hand thumb rule for direction of magnetic field

  • Curl the palm of your right hand around the circular wire with the fingers pointing in the direction of the current. The right-hand thumb gives the direction of the magnetic field.
• Ampere's circuital law

  ∮B·dl = μ0 * I, where I is the total current passing through the surface bounded by the closed loop
• Ampere's circuital law holds for steady currents which do not fluctuate with time
• Ampere's circuital law may not always facilitate an evaluation of the magnetic field in every case
• Solenoid


  Long wire wound in the form of a helix where the neighbouring turns are closely spaced
• Magnetic field inside a long solenoid is uniform, strong and along the axis of the solenoid
• Magnetic field outside a long solenoid is weak and along the axis of the solenoid with no perpendicular or normal component
• Finite solenoid


  • Magnetic field between two neighbouring turns vanishes
  • Magnetic field at interior mid-point P is uniform, strong and along the axis of the solenoid
  • Magnetic field at exterior mid-point Q is weak and along the axis of the solenoid with no perpendicular or normal component
• Long solenoid


  • Magnetic field outside the solenoid approaches zero
  • Magnetic field inside becomes everywhere parallel to the axis
• Determining magnetic field inside long solenoid

  1. Consider rectangular Amperian loop abcd

  2. Field along cd is zero

  3. Field components along bc and ad are zero

  4. Field along ab is B

  5. Use Ampere's circuital law to get B = μ0nI
• Magnitude of magnetic field inside the solenoid is 6.28 x 10^-3 T
• Parallel current-carrying conductors


  • Currents in same direction attract
  • Currents in opposite directions repel