PHYS 208 Lecture 16

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Biot-Savart Law

Equation for finding a magnetic field at any point ${\vec {r}}$ away from a current-carrying wire.

${\vec {B}}={\frac {\mu _{0}}{4\pi }}\int {\frac {I\mathrm {d} {\vec {\ell }}\times {\hat {r}}}{\left|{\vec {r}}\right|^{2}}}$

For an infinitely long, straight current carrying wire

${\vec {B}}(r)={\frac {\mu _{0}I}{2\pi r}}$

Alternate Right-Hand Rule

Thumb along current, fingers point in direction of magnetic field.

For oncoming current, magnetic field is always counter-clockwise

Force Between Parallel Wires

Two infinitely long wires are separated a distance $d$ with the currents $I_{1}$ and $I_{2}$ of both flowing in the same direction.

The magnetic field at each point along each wire point in opposing direction.

${\begin{aligned}{\vec {F}}&=I{\vec {\ell }}\times {\vec {B}}\\{\vec {B}}&={\frac {mu_{0}I}{2\pi d}}\\{\frac {\vec {F}}{\ell }}&=\mu _{0}{\frac {I_{1}I_{2}}{2\pi d}}\end{aligned}}$

• When the currents are parallel, the wires attract.
• When the currents are anti-parallel (in opposite direction), the wires repel.

Magnetic Field at Center of Current Loop

A circular loop of radius $R$ has a current $I$ flowing CCW. What is the magnetic field at the center of the circle?

(Dr. Webb refused to simplify his equation due to the symmetry, so I have taken the liberty to do it for him)

At every point around the circumference, the magnetic field points out from the loop towards us, so ${\vec {B}}$ is in the ${\hat {k}}$ direction.

${\begin{aligned}{\vec {B}}&={\frac {\mu _{0}}{4\pi }}\int {\frac {I\mathrm {d} \ell }{R^{2}}}{\hat {k}}\\&={\frac {\mu _{0}I}{4\pi R^{2}}}\int \mathrm {d} \ell {\hat {k}}\\&={\frac {\mu _{0}I}{4\pi R^{2}}}\left(2\pi R\right){\hat {k}}\\&={\frac {\mu _{0}I}{2R}}{\hat {k}}\end{aligned}}$

Now that was much easier to understand, involved fewer steps, and still arrived at the same answer.

For any fraction of a circle, ${\vec {B}}$ scales accordingly:

• Semicircle (half of a circle): ${\frac {1}{2}}{\vec {B}}={\frac {\mu _{0}I}{4R}}{\hat {k}}$
• Quarter circle: ${\frac {1}{4}}{\vec {B}}={\frac {\mu _{0}I}{8R}}{\hat {k}}$

Ampere's law

Finding magnetic fields using symmetry:

Suppose we integrate a magnetic field "flowing" through a closed wire loop:

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