Magnetic field uniformity problem

[Callix]

Homework Statement

While at an engineering internship, you reminisce about the similarities between gravitational fields, electric fields, and magnetic fields. You know that two infinite sheets of opposite charge can be used to create a uniform electric field (for points between the two sheets) and you know that two infinite arrays of opposite current carrying wires can produce a uniform magnetic field (for points between the two arrays), but the direction of your internship reminds you that there is another possibility. You recall that two large coils with N tight turns of radius R can yield a uniform magnetic field. If the two coils are separated by a distance R, where along the x-axis is the magnetic field uniform, and what additional calculations might you need in order to convince your director that the field is indeed uniform?

Homework Equations

The Attempt at a Solution

I found an image that depicts how I was visualizing this scenario

Wouldn't the field simply be uniform along the x-axis? Or would it be specifically at the center of each ring/coil. I guess since the field decreases with increasing distance, then it would seem logical that the field is uniform at the centers.

Numerically, couldn't we obtain values of the field as a function of position, plot them, and determine where the derivatives are 0? Implying that the field is unchanging and thus uniform?

 

[BvU]

Check out eqn (8) here . Separation R gives the most uniform behavior at the center.

 

[mfb]

Wouldn't the field simply be uniform along the x-axis?

Certainly not everywhere. 100 km away the field will be weaker than close to the coils.

Or would it be specifically at the center of each ring/coil.

Why do you expect a uniform field there? Is there some other special point where the field might be uniform?

Numerically, couldn't we obtain values of the field as a function of position, plot them, and determine where the derivatives are 0? Implying that the field is unchanging and thus uniform?

Sure.

 

[Callix]

Yeah, I was just kind of rambling through my thought process without going back to remove it.. XD

At the time I observed each ring and thought it would be uniform at the center of each ring, but then after I realized that the other ring will have an affect as well.

 

[Callix]

Is there some other special point where the field might be uniform?

So I imagine the the only uniform point would be in the center of the circular cross-section at a distance R/2.

 

[mfb]

Right.

 

[BvU]

So I imagine the the only uniform point would be in the center of the circular cross-section at a distance R/2

Link in post #2 shows very good uniformity from -R/3 to + R/3 (their R = 0.2 m)

Not clear to me where "the center of the circular cross-section at a distance R/2" is located...

 

[Callix]

Link in post #2 shows very good uniformity from -R/3 to + R/3 (their R = 0.2 m)

Not clear to me where "the center of the circular cross-section at a distance R/2" is located...