# Finding Quantitative Proof of Optimal Helmholtz Coil Separation

• locke
In summary: That should give you a better idea of how uniform the field is at that point.In summary, the most uniform magnetic field between two Helmholtz coils occurs when the separation between the coils is equal to their radii.
locke
I'm trying to track down a quantitative proof that the most uniform magnetic field between two Helmholtz coils occurs for a separation equal to their radii.

So far I've just been playing around with the Biot-Savart law and proving that B is identical at several trial points along the axis through the centre of the coils in this arrangement, but I can't figure out how to prove it for _every_ point along the axis.

Any help greatly appreciated.

locke said:
I'm trying to track down a quantitative proof that the most uniform magnetic field between two Helmholtz coils occurs for a separation equal to their radii.

So far I've just been playing around with the Biot-Savart law and proving that B is identical at several trial points along the axis through the centre of the coils in this arrangement, but I can't figure out how to prove it for _every_ point along the axis.

Any help greatly appreciated.

If you are Java enabled, I suggest you begin by playing with the applet at this site

http://webphysics.davidson.edu/faculty/dmb/Helmholtz/HelmholtzCoils.html

Be sure to read carefully. It is not very long. The "time" in the graph is not time in this case, it is the radius of the coils. It is a graph applet that was no doubt created for general use and re-used here. Click on the Start the Exercise to see how the field along the axis of the coils varies as the radius varies. It should occur to you that when the curve gets flat in the middle, the field is uniform, so if you had a way to quantify the flatness of the middle of the curve, you would have an approach to the problem.

The flatness of curve can be quantified by calculating the radius of curvature of the curve. If you don't know how to do that, look here, especially at equation #5.

What value of the radius of curvature represents the most uniform field? What radius of the Helmholtz coil produces that condition?

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Try expanding the expression for the field as a taylor series about the point halfway the center of the coils.

## 1. What is a Helmholtz coil?

A Helmholtz coil is a pair of identical circular coils placed in parallel and separated by a distance equal to their radius. When electric currents flow through the coils in the same direction, they produce a nearly uniform magnetic field between them.

## 2. Why is finding the optimal separation of Helmholtz coils important?

Optimal separation of Helmholtz coils is important because it determines the strength and uniformity of the magnetic field they produce. This is crucial for various scientific experiments and applications, such as in particle accelerators, magnetic resonance imaging (MRI) machines, and magnetic levitation systems.

## 3. How can one find the optimal separation of Helmholtz coils?

The optimal separation of Helmholtz coils can be found by measuring the magnetic field strength at different distances from the coils and plotting a graph. The point at which the field strength is most uniform and highest is considered to be the optimal separation.

## 4. What are the factors that affect the optimal separation of Helmholtz coils?

The optimal separation of Helmholtz coils is affected by the coils' radii, the distance between them, the number of turns, the current flowing through the coils, and the properties of the materials used to make the coils.

## 5. Can the optimal separation of Helmholtz coils be calculated mathematically?

Yes, the optimal separation of Helmholtz coils can be calculated mathematically using the following formula: d = (4/5)^(1/5) * R, where d is the optimal separation and R is the radius of the coils. However, this formula is only an approximation and may not always give the most accurate results.

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