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.

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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.

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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.

\[ d\vec{l}\times\hat{r} = (-\hat{x}) \times (-\hat{z}) = -\hat{y}\]

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 \((x_1, 0)?\)

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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?

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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.

1. 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