Magnetic Field in a Single Current Coil

[opsb]

I understand that the magnetic field in a solenoid can be approximated as being constant as the length of the solenoid tends to infinity, but I was wondering if anyone could show me or point me in the direction of a derivation of the precise magnetic field at any point within a single loop of current. The magnetic field at any point along an axis perpendicular to the coil running through its centre can be determined, is there a solution to the field at any point in the plane of the coil. I wrote the problem down and got an integral that looked pretty hard, and I couldn't figure out a way to solve it. Any help would be appreciated.
Thanks.

 

[cragar]

Did you try the biot-savart law ? It might work

 

[opsb]

I did, but you get this really tricky integral. I was wondering if there might be a smarter way to do it than brute force.

 

[torquil]

When you run into a difficult integral: just use it to define a function and name it after yourself!

Although in this case it seems to be elliptic integrals, which are already taken :-)

Did you get the same as this?:

http://www.netdenizen.com/emagnet/offaxis/iloopoffaxis.htm"

 

[sardar]

I would be happy to provide a response to your inquiry about the magnetic field in a single current coil. The magnetic field in a single current coil is a complex topic and has been studied extensively by researchers in the field of electromagnetism. There are several ways to approach this problem, and I will outline a few potential solutions for you to explore further.

One approach is to use Ampere's law, which relates the magnetic field to the current flowing through a closed loop. In the case of a single current coil, the closed loop can be taken as the perimeter of the coil. By integrating over this loop, you can determine the magnetic field at any point in the plane of the coil. This approach requires some mathematical rigor and may be the integral that you mentioned in your inquiry. However, it is a valid and widely used method for calculating the magnetic field in a single current coil.

Another approach is to use Biot-Savart's law, which relates the magnetic field to the current density at a given point. This method is more general and can be applied to any current-carrying wire or coil. By considering the current density at different points along the coil, you can determine the magnetic field at any point in the plane of the coil. This approach also requires some mathematical calculations, but it may be more intuitive and easier to understand than Ampere's law.

There are also computer simulations and numerical methods that can be used to model the magnetic field in a single current coil. These methods use mathematical algorithms and computer programs to solve the equations that describe the behavior of the magnetic field. This approach may be useful if you are interested in visualizing the magnetic field or if you want to explore different parameters, such as the shape or size of the coil.

In conclusion, there are several ways to approach the problem of determining the magnetic field in a single current coil. I recommend exploring each of these methods and seeing which one works best for your specific problem. Additionally, there are many resources available online and in scientific literature that provide derivations and solutions to this problem. I hope this response has been helpful, and I wish you the best of luck in your research.