### Creating a Uniform Magnetic Field

Located in Brookhaven National Laboratory in New York, this large Helmholtz coil is used to produce a uniform magnetic field.

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- A Helmholtz coil consists of a pair of identical circular magnetic coils that are separated by a distance equal to their radii. The magnetic field at a distance \begin{align*}x\end{align*}
x from the center of one coil is given as:

\begin{align*}B=\frac{N \mu_o I R^2}{2}\left[ \frac{1}{(R^2 + x^2)^{\frac{3}{2}}}+ \frac{1}{(2 R^2 - x^2 - 2 Rx)^{\frac{3}{2}}} \right]\end{align*}

Where \begin{align*}I\end{align*} is the current through the coils, \begin{align*}N\end{align*} is the number of turns in the coil and \begin{align*}R\end{align*} is the radius of both of the coils. By looking at the change in the magnetic field along the \begin{align*}x\end{align*}-axis, it can easily be shown that this setup creates a region of nearly uniform magnetic field.

- One popular application for a Helmholtz coil is to create a region of space with a uniform magnetic field. By creating this region, scientists are able to use this method to study the magnetic properties of matter.

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Using the information provided above, answer the following questions.

- If the current is increased in a Helmholtz coil, does the magnetic field change? How?
- Is the number of turns in the coils proportional or inversely proportional to the magnetic field?
- Show that at a point on the axis between the two coils, the magnetic field is given as:

\begin{align*}& N \mu_o IR^2 \left(\frac{1}{(\frac{5}{4}R^2)^{\frac{3}{2}}} \right) \\ B &=\frac{N \mu_o IR^2}{2} \left[\frac{1}{(R^2+x^2)^{\frac{3}{2}}}+\frac{1}{(2 R^2+x^2-2Rx)^{\frac{3}{2}}} \right] \\ \text{let} \ x &=\frac{1}{2}R \\ B &=\frac{N \mu_o IR^2}{2} \left[\frac{1}{\left[R^2+ \left(\frac{1}{2}R \right)^2 \right]^{\frac{3}{2}}}+\frac{1}{\left(2 R^2+\left(\frac{1}{2} R \right)^2-2R \left(\frac{1}{2} R \right) \right)^{\frac{3}{2}}} \right] \end{align*}