Magnetic Field

• electric currents create magnetic fields.
• units are teslas

Right Hand Rule §

3 Properties §

1. covers all points around current carrying wire
2. vector
3. exerts forces on magnetic poles

Magnetic Fields from Moving Charges §

Biot-Savart Law §

• basically Coulomb’s law for $\vec{B}$
• ${\mu }_0$ is permeability constant

$${\vec{B}}_{\text{point charge}}=\frac{{\mu }_{0}qv\sin \theta }{4\pi r^2}$$

Magnetic Field of a Current §

Biot-Savart Law §

Using the Biot-Savart Law for a moving charge, we rewrite it in terms of many moving charges (current).

$$\begin{array}{rl}\vec{B}& =\int \frac{{\mu }_{0}v\sin \theta }{4\pi r^2}dq\\ & =\int \frac{{\mu }_0\sin \theta }{4\pi r^2}\frac{dl}{dt}dq\\ & =\int \frac{{\mu }_0\sin \theta }{4\pi r^2}Idl\\ & =\frac{{\mu }_0}{4\pi }I\int \frac{x}{r}\cdot \frac{1}{r^2}dl\\ & =\frac{{\mu }_0}{4\pi }I\int \frac{x}{\sqrt{x^2+l^2}}\cdot \frac{1}{x^2+l^2}dl\\ & =\frac{{\mu }_0}{4\pi }Ix{\int }_{-\infty }^{\infty }\frac{1}{(x^2+l^2{)}^{\frac{3}{2}}}dl\\ & =\frac{{\mu }_{0}Ix}{4\pi }\end{array}$$