Ampere's Law: Moving Electric Charges Generate a Magnetic Field

Ampere's Law

$\int\vec{B}d\vec{l} = \mu_oI_c$ $B$ = magnetic field ( $T$ )

$dl$ = length of surface ( $m$ )

$\mu_o$ = $4\pi$ x $10^{-7}$

$I_c$ = current through the closed surface ( $A$ )

Application of $\vec{B}$ :

Magnetic field OUTSIDE a wire:

\int\vec{B}d\vec{l} = \mu_oI_c

\vec{B}\int d\vec{l} = \mu_oI_c

\vec{B}(2\pi r) = \mu_oI_c

\vec{B} = \frac{\mu_oI_c}{2\pi r}

Magnetic field INSIDE a wire:

\vec{B}(2\pi r) = \mu_oI_c\frac{V_r}{V_R}

\vec{B}(2\pi r) = \mu_oI_c\frac{\pi r^2L}{\pi R^2L}

\vec{B} = \frac{\mu_oI_cr}{2\pi R^2}

Magnetic Field Inside a Toroidal Coil:

\int\vec{B}d\vec{l} = \mu_oI

\vec{B}(2\pi r) = \mu_oNI

\vec{B} = \frac{\mu_oNI}{2\pi r}

$N$ = number of turns of coil

Note:

Test Yourself!

What is the magnetic field inside a long, straight wire at a distance of 15 $cm$ from its center. The diameter of the wire is 50 $cm$ . The wire carries a current of 5.0 $A$

The magnetic field at a distance of 50 $cm$ from a long, straight wire carrying a current of 5.0 $A$ is approximately

Challenge yourself!

What is the magnitude of the magnetic field 25% of the way between wire A and wire B (at point P). The current in each wire is 0.6A. The distance between the wires is 40cm.