Generalise the magnetic field of multiple current loops

[Orlando]

Homework Statement

Determine an equation to calculate the flux density at the center of a range of current loops encircling each other.

Homework Equations

The equation for the magnetic flux density at the center of a current loop is

mu*I / (2*r)

The Attempt at a Solution

I assumed I could just integrate the above equation over a range of radii as follows:

b
\int(mu*I / (2*r)) dr = (mu*I/2)*-ln(a/b)
a

Have I done this correctly or do the magnetic fields not add this way? Please help!

PS. How on Earth do you use the maths tools to write equations in these posts.

 

[Borek]

PS. How on Earth do you use the maths tools to write equations in these posts.

https://www.physicsforums.com/showthread.php?t=8997

 

[aim1732]

Besides dimensional incorrectness you should realize that the current that appears in the expression is [n*i] i.e the number of turns per unit length times the length. What you can do is write the current as n*dr where dr is differential radius.Then you integrate.You realize now that you can not expect integral of field times length to give you field.

 

[Orlando]

Thank you for your response. I must apologise - I've never really learned how to use calculus in this manner and still don't quite understand. If it's not too much effort, could you write out the integral? I have run out of time and don't understand how to manipulate integrals. It would be much appreciated.

 

[aim1732]

To write out any integral you first need to write your basic equation--which in this case is the B field at the centre of the loop--for a differential element.That element could be a small ring(differential radius),length(differential length),current element,mass element,etc.

In the formula for B field at the centre of the field the current appearing is actually differential--you would appreciate that it is only a small part of the total current.But the variable in your equation is radius r.So you manipulate element di as n*dr where n is the number of current loops per unit radius which in this case is N/b-a[N--total loops].Of course uniform distributions of loops is assumed.

Now you integrate accordingly.It can be easy if you always remember that the element of integration should come out of the basic equation.If you are adding it later it will be almost always wrong.

 

[Orlando]

Ah, I understand it much better now. Thank you very much for your help, I appreciate it.