Magnetic field

<p>There is a small green magnifying glass initially at the origin that indicates where the magnetic field is measured. The measurement point can be moved in three dimensions by adjusting the components of its position vector $\vec{r}$. </p>

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<p>The magnetic field shown above is due to a short wire segment with a current flowing through it in the $x-$direction. The magnetic field was calculated with the Biot-Savart law,</p>

$$d\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\frac{I d\vec{r}_{wire} \times (\vec{r}-\vec{r}_{wire})}{|\vec{r}-\vec{r}_{wire}|^3}\quad\text{[T]}.$$

<p>Here position $\vec{r}$ is the position where the magnetic field is calculated, $\vec{r}_{wire}$ is the position of the center of the wire segment, the vector  $d\vec{r}_{wire}$ has the length of the wire segment and points in the direction that the current is flowing, and $\mu_0 = 4\pi \times 10^{-7}$ T m/A is the permeability constant. You might have to zoom in to see the wire segment at the origin. </p>

<p>An isolated wire segment with a current flowing through is unphysical because the current cannot just appear at one end of the wire and disappear at the other end. However, to calculate the magnetic field due to a current flowing in a long wire, the wire is typically divided into short segments like this and the contribution from each segment is summed up.</p>

<p>The magnetic field produced by an infinitely long straight wire is,</p>

$$\vec{B}(\vec{r})=\frac{\mu_0I}{2\pi|\vec{d}_{\perp}|^2}\,\hat{n}\times\vec{d}_{\perp}\,\,\text{[T]},$$

<p>where $I$ is the current flowing through the wire, $\hat{n}$ is a unit vector pointed in the direction that the current is flowing, $\vec{d}_{\perp}$ is the shortest vector pointing from the wire to the measurement point $\vec{r}$, and $\mu_0$ is the permeability constant.</p>

<p>Change the direction that the current flows and notice that the magnetic field always points in the direction of the right-hand rule. Point your thumb of your right hand in the direction that the current is flowing and curve the fingers of your right hand. Your fingers will point in the direction of the magnetic field.</p>

<p>Consider a wire loop of radius $R$ in the $x-y$ plane and centered at the origin. The <a href="http://lampz.tugraz.at/~hadley/physikm/script/magnetism/loop.en.php">magnetic field along the $z$-axis</a> is,</p>

$$\vec{B}(z)=\frac{\mu_0I  R^2\hat{z}}{2\sqrt{R^2+z^2}^3}\hspace{1cm}\text{[T]}.$$

<p>In the example shown above, $R=1$, $I=1$ so at $z=1$ the magnetic field should be $2.22\times 10^{-7}$ T. This can be checked by  measuring the magnetic field at that point. There is no simple formula for the magnetic field at positions not on the $z-$axis, but it can be measured using the <input type="button" value="Measure B at position r" onclick="document.getElementById('txt1').innerHTML=document.getElementById('calculate').value;  MathJax.Hub.Queue(['Typeset',MathJax.Hub]); field();"> button..</p>

<p>Consider two concentric wire loops with radii $R_1$  and $R_2$ in the $x-y$ plane and centered at the origin. The <a href="http://lampz.tugraz.at/~hadley/physikm/script/magnetism/loop.en.php">magnetic field along the $z$-axis</a> is,</p>

$$\vec{B}(z)=\frac{\mu_0I_1  R_1^2\hat{z}}{2\sqrt{R_1^2+z^2}^3}+\frac{\mu_0I_2  R_2^2\hat{z}}{2\sqrt{R_2^2+z^2}^3}\hspace{1cm}\text{[T]}.$$

<p>There is no simple formula for the magnetic field at positions not on the $z-$axis, but it can be measured using the <input type="button" value="Measure B at position r" onclick="document.getElementById('txt1').innerHTML=document.getElementById('calculate').value;  MathJax.Hub.Queue(['Typeset',MathJax.Hub]); field();"> button.</p>

<p>The top view for parallel currents seems not circularly symmetric. This comes from the discrete spacing of the arrows. Use the zoom and pan buttons to investigate the magnetic field near the wires to see that the pattern is circularly symmetric. </p>

<p>This is the magnetic field produced by a 6 turn helical coil. Try looking at the top view and the side view while changing the +z and -z buttons to see the form of the magnetic field.</p>

<p>The formula for the magnetic field in a very long coil with many turns is $B = \mu_0 nI$ where $n$ is the number of turns per meter. In this example $n=10$ and the formula predicts a magnetic field of $B=7.5\times 10^{-6}$ T. The magnetic field in this coil is about half of that value. This coil is not long compared to its radius and the spacing between the winding is not much smaller than the radius.</p>

<p>The image below is zoomed in close to the edge of the coil.</p>

<p>The image below illustrates a three dimensional magnetic field created by a collection of wires with currents flowing through them. The magnitude of the magnetic field at every point is given in opacity of the vectors. The black vectors have the largest magnitude and the transparent vectors have the smallest magnitude. The arrows show the components of the magnetic field in the $x−y$ plane. If the arrow is short and black, that vector is pointing mostly out of the $x−y$ plane. If the vector is pointing out of the screen, $\odot$ is displayed. If the vector is pointing into the screen, $\otimes$ is displayed.  It is possible to adjust the $z-$plane that is displayed by pressing the <input type="button" value="z+"> and <input type="button" value="z-"> buttons.</p>

<p>Either infinitely long straight wires or curved wires can be added to the collection with the <input type="button" value="+"> buttons and the last one can be removed with the <input type="button" value="-"> button. By pressing on the numbered buttons, you can change the orientation of a wire and the current flowing through it. If a wire is added outside the field of view, press the <input type="button" value="rescale"> button. </p>

<p>When the <input type="button" value="Measure B at position r"> button is pressed, a small green magnifying glass is displayed that can be moved around to measure the magnetic field at any position.</p>

<p>To view a 3-D drawing of the wires that have been defined, press the <input type="button" value="show"> button. This drawing is hidden when the <input type="button" value="hide"> button is pressed. There is a algorithm that tries to choose the right scaling for the opacity of the magnetic field vectors but  algorithm sometimes fails because the formulas for the field diverge at the wires and you have to adjust the opacity manually.</p>

<p>There is a pulldown menu that will load various configurations.</p>

Falstad: 3-D Magnetostatic Field Simulation






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