Magnetic Force

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Discussion

introduction

Magnetism is the force that moving charges exert on one another. This formal definition is based on this simple formula.

FB = qv × B

Recall that electricity is (in essence) the force that charges exert on one another. Since this force exists whether or not the charges are moving, it is sometimes called the electrostatic force. Magnetism could be said to be an electrodynamic force, but it rarely is. The combination of electric and magnetic forces on a charged object is known as the Lorentz force.

F = q(E + v × B)

For large amounts of charge …

FB = q  v × B
   
FB = q  dx  × B =  dq ℓ × B
dt dt
   
FB = I  ℓ × B

This formula for the magnetic force on a current carrying wire is the basis for the experiment defining the fundamental unit of electric current in the SI system, the ampère.

The ampère is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one meter apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per meter of length. (BIPM 1948)

Using Ampère's law, we derived a formula for the strength of the magnetic field surrounding a long straight current carrying wire …

B =  μ0I
2πr

Substitute this expression into the magnetic force formula. (Since the two wires are parallel the field of one strikes the other at a right angle and the cross product reduces to straight multiplication.) The solve for the force per unit length as described in the experiment …

FB  =  Iℓ × B
       
FB  =  Iℓ μ0I
2πr
       
FB  =  μ0I2
ℓ 2πr

This sets the permeability of free space to its unusually precise value (unusually precise for a physical constant). Substitute the values for the measurements described in the BIPM experiment into the last equation we derived …

FB  =  μ0I2
ℓ 2πr
     
(2 × 10−7 N)  =  μ0 (1 A)2
(1 m) 2π (1 m)

and solve for the permeability of free space …

μ0 =  2π (1 m)(2 × 10−7 N)
(1 m)(1 A)2
   
μ0 =  4π × 10−7 N/A2

Returning to formula for the magnetic force on a current carrying wire leads to the following definition of magnetic field strength and its unit, the tesla.

FB = I ℓ × B ⇒ B =  FB ⇒ ⎡
⎣		 T =  N ⎤
⎦
Iℓ Am

Third right/left hand rule


[magnify]

Cathode ray tube: color television (color monitor), oscilloscope,


[magnify]

mass spectrometer

cyclotron

space weather, aurora, van allen radiation belts

electric motor

electromagnetic rail gun

nuclear magnetic resonance?

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something.
    • Answer it.
  3. Write something.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. How does the direction of a magnetic force on a moving charged particle differ from the direction of an electric force? State the direction of each force relative to the respective field.
  2. Show the direction of the force acting on the current-carrying wire between the two bar magnets.
     
     
  3. Show the direction of the force acting on the electron moving between the poles of a horseshoe magnet.
     
     
  4. The tracks highlighted in the photograph below were made by subatomic particles in a collider. The tracks highlighted in red and green were made by electrons.
     
     
    1. How are the tracks made by the electrons different from the other tracks? Why are they different?
    2. How is the track highlighted in red different from the track highlighted in green? Why is it different?
  5. A positively charged particle is placed at rest in a region of uniform electric and magnetic fields. Describe …
    1. the direction of the electric and magnetic forces just as soon the particle gets moving and
    2. the resulting path of the charge
    if the two fields are …
             
       
    1. parallel
    		 
    1. antiparallel
    		 
    1. perpendicular
             

algebraic

  1. Three problems about electromagnetic units …
    1. Show that the form used on most reference tables for the permeability of free space (μ0 = 4π × 10−7 Tm/A) is equivalent to the form derived in the discussion above (μ0 = 4π × 10−7 N/A2).
    2. Express the permeability of free space in terms of funamental units.
    3. Reduce the tesla to its equivalent in fundamental units.

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