Biot-Savart Law

 Drawing the magnetic field: field vectors vs field lines

 The source of the magnetic field:  Moving charges

• The current in the wire consists of moving charges

• A moving charge alters the space to produce a magnetic field.

• A stationary charge will not produce a magnetic field

The Biot-Savart Law:  The magnetic field strength at a point due to a moving charge

B = (u0/4π)(qvsinθ/r2)

θ – angle from velocity (v) to line (r) between charge and point of interest.
 

The SI unit is the Tesla

1T = 1 N/(Am)

u0 = 4π x 10-7 Tm/A (permeability constant)

Biot-Savart Law:  Direction

The direction of the field is given with the right hand rule, with thumb pointing in direction of charge motion and fingers showing orientation and direction of magnetic field.

 B is zero along the line of charge motion (θ = 00 or 1800)

Biot-Savart Law:  Vector Cross Product

The Biot-Savart Law can be written in terms of a cross-product:

 

Note the r in the numerator is the unit vector; it has a value of 1 (don’t divide i out of the denominator!) and direction of r.

  

Alternate way of using a RH rule to determine direction of cross-product

 

Numerical Problem

What are the magnetic field strength and direction at the dot in the figure?

Note axes in 10-2 m

Problem solving tip:

u0/4π = 1 x 10-7 Tm/A    

 Answer: Numerical Problem Sin θ and RH rule method

r = .02      m, θ = 1350

B = (u0/4π)(qvsinθ/r2)

Answer:  2.83 x 10-16 T

Direction: out of the page.

 Answer: Cross product method

v = 2 x 10-7 j

r = (- .02 i - .02 j)

   = (- i - j)

B = (u0q/4πr2)v(j)X(-i-j)

=(u0qvsinθ/4πr2)(j)X(-i-j)

But since j X ±j = 0

B = =(u0qvsinθ/4πr2)(j)X(-i)

j X –i = +k

Answer:  2.83 x 10-16 T k