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:

$\oint \overrightarrow{B}\cdot \mathrm{d}\overrightarrow{\ell }={\mu }_0{I}_{en}$

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):

$B\cdot L=\oint \overrightarrow{B}\cdot \mathrm{d}\overrightarrow{\ell }={\mu }_0(I_{out}-I_{in})$

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:

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

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

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

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