Magnetic Induction: Solving 3 Formulas

[Andrea]

Hello everybody and thanks so much for your help,
(I know I have to improve my English)

here is my big problem concerning 3 formula's :
(It's just the beginning)
-first at all,we have a magnetic field B created by the circulation of a current I into a spire (with a spoke a) on a point at a distance b of the plane of the spire on his axis
-->B(b) = µo I a²/ 2(a²+b²)^3/2
-then,we have an other magnetic field with 2 identical coils separated with a distance 2b just to be in the same axis and to have the same current I circulating in the same direction.
-->B(x)=B1(x) + B2(x) = µo I a²/2 ((1/(a²+b+x)²)^3/2) + (1/(a²+(b-x)²)3^/2)).
-It's possible to give a Taylor's development of this expression,so we obtain the Helmholtz condition (I don't understand that very well ) and the magnetic field is given by:BH=8µ0NI/5 (square root 5)a.

Now I have to find these 3 expressions:
aid of the Biot-Savart law,how could you obtain the expression of the magnetic field created by a current I circulating into a spire on a point of the normal axis in the plane of the spire and passing through his centre.
>>From this expression,how could you deduce the magnetic field on any point of this axis of an Helmholtz assembly.You have to give the taylor's development in the powers of x around the centre of the assembly and deduce the Helmholtz condition 2b=a which verifyes an homogeneous magnetic field in the assembly.Which is the value of the magnetic field?At which distance of the centre,on the axis of the assembly,is the field different of 1% from his value at the centre?
(*concerning the assembly,we use a Hall Effect probe too)

Please help me because I don't understand anything.I am desperate .What is the Helmholtz condition?Why is that so important?Could you please help me for the questions (principally from>>>)

Thank you very much!

 

[Doc Al]

Field in a Helmholtz Coil

What is the Helmholtz condition?Why is that so important?

Welcome to Physics Forums!

A Helmholtz coil uses two current-carrying coils to create a fairly uniform field at the center. The basic idea of your problem is to calculate the separation distance between the coils that will give the most uniform field along the axis. The field at the center is, of course, the superposition of the field from each coil. By doing a Taylor series expansion of the field, you can find the separation distance that will give you the most linear field: the idea is to find the separation distance that will make 2nd order terms cancel. That separation distance is the so-called Helmholtz condition: for circular loops I think the distance between the coils should equal the radius of the coils for maximum uniformity.

I hope this helps a little.

 

[Andrea]

...

Thank you...
Concretely,I need to solve the questions and I'm not a genius in arithmetic So I have understood the principle but It seemed to be developed.
"From this expression,how could you deduce the magnetic field on any point of this axis of an Helmholtz assembly.You have to give the taylor's development in the powers of x around the centre of the assembly and deduce the Helmholtz condition 2b=a which verifyes an homogeneous magnetic field in the assembly.Which is the value of the magnetic field?At which distance of the centre,on the axis of the assembly,is the field different of 1% from his value at the centre?"
I'm sincerely sorry but I don't know how to use the formula's.Is it possible for someone to give me a little arithmetic development?

Thank you (2 x)!

 

[Doc Al]

I'm still not sure what's causing you trouble, but let me give a bit of explanation of each equation. Maybe that will help.

-first at all,we have a magnetic field B created by the circulation of a current I into a spire (with a spoke a) on a point at a distance b of the plane of the spire on his axis
-->B(b) = µo I a²/ 2(a²+b²)^3/2

This equation comes directly from the law of Biot-Savart expressed for current. If you are not familiar with this, look it up! It tells how to find the magnetic field from a section of current-carrying wire. In this case you want to find the axial field of a coil: you would integrate the current elements around the coil to get the magnetic field along the axis. Since you just need the axial field, it's not that difficult.

In this equation, "a" is the radius of the coil and "b" is the distance along the axis from the center of the coil.

-then,we have an other magnetic field with 2 identical coils separated with a distance 2b just to be in the same axis and to have the same current I circulating in the same direction.
-->B(x)=B1(x) + B2(x) = µo I a²/2 ((1/(a²+b+x)²)^3/2) + (1/(a²+(b-x)²)3^/2)).

This is just the sum of the fields from two coils. You have a typo, the correct equation is:
B(x)=B1(x) + B2(x) = µo I a²/2 ((1/(a²+(b+x)²)^3/2) + (1/(a²+(b-x)²)3^/2)).
In this equation, "b" is the distance from each coil to the midpoint between the coils. "x" is the position along the axis measured from that midpoint.

-It's possible to give a Taylor's development of this expression,so we obtain the Helmholtz condition (I don't understand that very well ) and the magnetic field is given by:BH=8µ0NI/5 (square root 5)a.

If you don't understand the Taylor series expansion, then look it up! (You'll have to take derivatives of that expression for B(x).)

The basic idea is that the field can be represented by a Taylor series expansion about the midpoint. Each term represents the deviation of the field from uniformity. The Helmholtz condition will make the 2nd order term vanish, making the field relatively uniform. (That condition turns out to be 2b = a.) So start cranking!

Assuming you have shown that 2b = a, then just plug that into find the zeroth term of the field at the midpoint (x = 0).

To find variation from that term, evaluate the first non-zero term in the Taylor expansion.