PHYS 208 Lecture 17

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Superposition of Magnetic Fields

The magnetic field of a point in the vicinity of multiple current-carrying wires is the vector sum of the magnetic fields produced by each wire.

Ampere's Law

Integrating counter-clockwise around a closed loop (of any shape) surrounding a current-carrying wire (loop does not even have to be in perpendicular plane) always yields the same result:

$\oint {\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu _{0}I_{en}$

Even when there are multiple current-carrying wires, the $I_{en}$ is always the net sum of all the currents (sign of each $I$ results from CCW integration):

$B\cdot L=\oint {\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu _{0}(I_{out}-I_{in})$

where $L$ is the perimiter of the loop. The first form only works when $B$ is uniform at every point along the loop.

Pop Quiz

Suppose we have a thick current-carrying wire with current $I$ evenly distributed across the cross-sectional area. Taking the integral of a loop within the wire gives:

$\oint {\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu _{0}I\left({\frac {r}{R}}\right)^{2}$

Therefore, we can find the magnetic field of any radius $r$:

${\begin{aligned}B(r)\cdot (2\pi r)&=\oint {\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu _{0}I{\frac {r^{2}}{R^{2}}}\\B(r)&={\begin{cases}{\frac {\mu _{0}Ir}{2\pi R^{2}}}&r<R\\{\frac {\mu _{0}I}{2\pi r}}&r>R\end{cases}}\end{aligned}}$

Geometries

Ampere's law only works nicely for

• Long, straight wires
• Solenoids (wires wrapped around a cylinder): $B=\mu _{0}nI$, where $n$ is the number of coils per unit length
• Toroid (wires wrapped around a donut)
• Sheets of current