# Biot Savart Law with Different Magnetic Permeabilities

• BrandonBerchtold
In summary, the conversation discusses a program that generates a 2D heat map of the magnetic field produced by a finite length solenoid. The plan is to divide the conductor into small lengths and use Biot Savart Law to calculate the total field contributions from each length. The question is raised about the effect of adding a core to the solenoid and whether Biot Savart Law accounts for the increased field strength outside the core. A solution is suggested using the surface current model of magnetism or the pole method for computing the magnetic field.
BrandonBerchtold
I would like to make a program that produces a 2D heat map showing the magnitude of the magnetic field produced by a finite length solenoid. The heat map would show the field strength along the radial and axial directions of the solenoid.

I plan to divide the conductor into "infinitessimally" small lengths (on the order of 1-10 microns long), and sum up the total field contributions from each of these lengths for every axial and radial location along the coil to generate the 2D heat map, using Biot Savart Law.

My question is as follows: if a core is added to the solenoid, the field strength increases significantly, even outside of the core. Does Biot Savart Law account for this when the field is being calculated outside of the core? If I am summing the infinitessimally small field contributions of each current element, how would the locations outside of the core know that the field strength has increased?

Is there a way to deal with this problem similarly to how magnetic circuits are solved in transformers with air gaps, as they are still able to solve for the magnetic flux as it passes through both an air gap and the ferrite core.

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In the surface current model of magnetism, there are surface currents on the cylindrical iron core with the exact same geometry as the current in the solenoid. Biot-Savart can be used to compute the magnetic field from these surface currents. In general for a solenoid with current per unit length ## K=nI ## where ## n ## is the number of turns per unit length, the surface current per unit length is ## \mu_r-1 ## times that of the surface current per unit length ## K ## of the solenoid, where ## \mu_r ## is the relative magnetic permeability of the core material.
Alternatively, the pole method for computing the magnetic field gets the exact same answer, and is much simpler to compute than a Biot-Savart calculation on the surface currents.

BrandonBerchtold

## 1. What is the Biot Savart Law?

The Biot Savart Law is a fundamental law in electromagnetism that describes the magnetic field produced by a steady current. It states that the magnetic field at a point in space is directly proportional to the current, the length of the current, and the sine of the angle between the current and the observer's position.

## 2. How does the Biot Savart Law apply to different magnetic permeabilities?

The Biot Savart Law applies to different magnetic permeabilities by taking into account the material properties of the medium in which the current is flowing. The magnetic permeability of a material is a measure of how easily it can be magnetized, and it affects the strength of the magnetic field produced by a current.

## 3. What is the significance of different magnetic permeabilities in the Biot Savart Law?

The significance of different magnetic permeabilities in the Biot Savart Law is that it allows for a more accurate calculation of the magnetic field in different materials. Different materials have different magnetic permeabilities, which means that the same current can produce different magnetic fields depending on the material it is passing through.

## 4. How does the Biot Savart Law with different magnetic permeabilities impact real-world applications?

The Biot Savart Law with different magnetic permeabilities has many real-world applications, such as in the design of electromagnets, motors, and generators. It is also used in medical imaging techniques such as magnetic resonance imaging (MRI) and in the study of Earth's magnetic field.

## 5. Are there any limitations to the Biot Savart Law with different magnetic permeabilities?

One limitation of the Biot Savart Law with different magnetic permeabilities is that it assumes a steady current, which may not always be the case in real-world situations. It also does not take into account the effects of magnetic materials, such as ferromagnetic materials, which can significantly alter the magnetic field. Additionally, the Biot Savart Law is a classical theory and does not fully explain the behavior of magnetic fields at the quantum level.

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