Biot-Savart Law
The Biot-Savart law provides the definition for differential magnetic field, $d\vec{B},$ created when a current, $i,$ flows through an infinitesimal length of wire, $d\vec{l},$ at a distance, $r,$ away.
$d\vec{B} = \frac{\mu_0\text{ } i \text{ } d\vec{l}\times\hat{r} }{4\pi r^2}$
The Biot-Savart law is necessary to find the direction of a magnetic field due to a current and very handy for calculating the magnetic fields of different wire configurations.
Direction and the right hand rule
The direction of the magnetic field comes from $d\vec{l}\times\hat{r}$ and cross product properties.
$d\vec{l}$ is the differential length of wire. Its direction is the same as the direction of the current flowing through it.
$\hat{r}$ is a unit vector that points from the wire to the point of investigation.
A current flows in the negative x-direction through a piece of wire of infinitesimal length located at the origin. Find the direction of the magnetic field at a point on the negative z-axis.
In order to determine the direction, first write down $d\vec{l}$ and $\hat{r}.$ Since the current flows in the negative x-direction,
$d\vec{l} = -\hat{x}.$
The wire is located at the origin and the point of investigation is on the negtaive z-axis, so
$\hat{r} = -\hat{z}.$
Evaluate the cross product.
$d\vec{l}\times\hat{r} = (-\hat{x}) \times (-\hat{z}) = -\hat{y}$
Magnitude
When actually calculating the magnetic field, it is helpful to follow these general steps.
1) Evaluate $d\vec{l}\times\hat{r}.$
2) Change variables.
3) Integrate.
A rod of length $L$ lies on the $y$-axis with current $i$ running in the $+y$-direction and ends at $y_1$ and $y_2.$ What is the magnetic field at $(x_1, 0)?$
1) Evaluate $d\vec{l}\times\hat{r}.$
Since the current runs up the y axis, $d\vec{l} = d\hat{y}.$
$d\vec{l}\times\hat{r} = dy ||\hat{r}|| \sin\theta \hat{z} = dy \sin\theta \hat{z}$
2) Change variables.
Since $\theta$ is the angle between y-axis and the displacement vector pointing from the y-axis to $(x_1,0),$
$\sin\theta = \frac{x_1}{r}$
Hence,
$r = \frac{x_1}{\sin\theta}.$
Also,
$\tan\theta = \frac{x_1}{y}$
$y= \frac{x_1}{\tan\theta} = x_1 \cot\theta$
$dy = x_1 (-\csc^2\theta) d\theta = -x_1 \csc^2\theta d\theta$
So the Biot-Savart law currently looks like
$d\vec{B} = \frac{\mu_0\text{ } i \text{ } d\vec{l}\times\hat{r} }{4\pi r^2} = \frac{\mu_0 \text{ }i\text{ } dy \sin\theta \hat{z}}{4\pi \big(\frac{x_1}{\sin\theta}\big)^2} = \frac{\mu_0 \text{ }i\text{ } (-x_1 \csc^2\theta d\theta) \sin^3\theta \hat{z}}{4\pi x_1^2}$
$d\vec{B} = -\frac{\mu_0 \text{ }i\text{ } \sin\theta d\theta }{4\pi x_1} \hat{z}$
3) Integrate.
$\vec{B} = - \frac{\mu_0 i}{4\pi x_1} \hat{z} \int_{\theta_1}^{\theta_2} \sin\theta d\theta = \frac{\mu_0 i}{4\pi x_1} ( \cos \theta_2 - \cos \theta_1) \hat{z}$
$\cos\theta = \frac{y}{r}=\frac{y}{\sqrt{y^2+r^2}}$
What is the magnetic field if the rod in the example above has infinite length?
Start with
$\vec{B} = \frac{\mu_0 i}{4\pi x_1} ( \cos \theta_2 - \cos \theta_1) \hat{z}.$
As the length approaches infinity, $\theta_2 \rightarrow 0^\circ$ and $\theta_1 \rightarrow 180^\circ,$ so
$\vec{B} = \frac{\mu_0 i}{4\pi x_1} ( \cos 0^\circ - \cos 180^\circ) \hat{z} = \frac{\mu_0 i}{2\pi x_1}.$
This should look familiar to those who have studied magnetic fields.
References
- Vandegrift, G. Magnetic field element (Biot-Savart Law). Retrieved June 8, 2016, from https://commons.wikimedia.org/wiki/File:Magnetic_field_element_(Biot-Savart_Law).jpg