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# The Law of Biot-Savart

"A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective"
Edward Teller

- We have seen that a magnetic field exerts a force on a moving charge or current, just as an electric field exerts a force on a charge.

and

- In the electric case Coulomb's Law gives us a method of determining E from an arbitrary distribution of charges. For a point charge we have

> where is a unit vector from the point charge to location at which the E field is to be determined.

- The Biot-Savart Law provides a general method of determining the B field from an arbitrary current distribution.

> where μ0 is the permeability of the vacuum (free space) = 4π x 10-7 T.m/A, dl is a "current element" directed along the current in the wire and is a unit vector from dl to where the B field is to be calculated as in the diagram at right.

- There are a number of important points to be aware of:

> The law must be written in differential form since there is no such thing as a "point current". Note that a moving point charge is equivalent to a current and a stationary point charge does not create a magnetic field.
> Due to the cross product dB is at right angles to both dl and . You can't avoid considering 3 dimensions.
> The "sense" of dB (into or out of the plane in the above diagram) is determined by the basic definition of a vector cross product or equivalently another "right-hand-rule" shown at right.
> (μ0/4π) plays a similar role to the Coulomb constant, k (= 1/4πε0). But compare the values of μ0 and k; this is an indication of the relative strengths of the electric and magnetic interactions of charges.
> Similar to our initial statement of Coulomb's Law, the above expression of the Biot-Savart Law gives dB due to a current element in a vacuum. For real world applications wheres the current is not in a vacuum a slight adjustment must be made to take into account the magnetic properties of the medium. This can be done in a similar manner to the adjustment of Coulomb's Law in a dielectric medium, but is beyond the scope of this course.

- Needless to say determining the magnetic field is typically significantly more complicated than the electric field.

Energizer bunny arrested. Charged with battery.

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu