Magnetic Force

Discussion

introduction

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

FB = q v × 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  × B
dt dt
   
FB = I   × 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
r

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  =  I × B
       
FB  =  I μ0I
r
       
FB  =  μ0I2
r

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
r
     
(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.

dFB = I d × B  ⇒  B =  FB  ⇒ 
T =  N
I Am

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