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.
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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