Full Width Half Maximum of Single Coil

[mitchy16]

Homework Statement

Determine the Full Width Half Maximum (FWHM) using the single coil equation and fitting parameters. The FWHM is the measure of how broad (wide) a signal is before it loses half of its intensity. In this case, its a measure of the distance spanned before the magnetic field magnitude is halved.

Homework Equations

radius of loop (R) : 10.5cm = 0.105m
distance from point charge (z) : 20cm = 0.20cm
magnetic field strength (B) = 0.1366 mT
N, number of loops: only did with 1 coil, so I assume 1?
I, current: 2 A

The Attempt at a Solution

I do not quite understand what it means by fitting paramaters? Is that in reference to the radius, current and other values? I don't understand what I'm supposed to solve for. Any guidance is appreciated, thank you!

NOTE/EDIT: This question is not homework, it is for a lab, I just need guidance on how to do it.

 

[kuruman]

I do not quite understand what it means by fitting paramaters? Is that in reference to the radius, current and other values?

Yes.

I don't understand what I'm supposed to solve for.

Make a plot of B vs. z. Find values z₁ and z₂ such that B(z₁) = B(z₂) = (½)B_max.
See https://en.wikipedia.org/wiki/Full_width_at_half_maximum

 

[Merlin3189]

I'm only a bewildered spectator here, drawn by the normally simple idea (in other contexts) of FWHM.

But this looks almost like the axial field of a simple coil, whose max field is at the centre of the coil and falls off monotonically with distance. Meaning there can be only 1 value of | z | = 0.5 B_max. So unless this is a contorted way of asking for that single value of |z| or asking how that value of |z| varies with R, I'm perplexed.

What I'd like to ask, to perhaps clarify the situation for me, is what is the significance / meaning of the z 'hat' symbol ?

Edit: If it turns out to be a unit vector in the z direction, then ok. (sorry!)

 

[mitchy16]

I'm only a bewildered spectator here, drawn by the normally simple idea (in other contexts) of FWHM.

But this looks almost like the axial field of a simple coil, whose max field is at the centre of the coil and falls off monotonically with distance. Meaning there can be only 1 value of | z | = 0.5 B_max. So unless this is a contorted way of asking for that single value of |z| or asking how that value of |z| varies with R, I'm perplexed.

What I'd like to ask, to perhaps clarify the situation for me, is what is the significance / meaning of the z 'hat' symbol ?

Edit: If it turns out to be a unit vector in the z direction, then ok. (sorry!)

According to the my professor, the z hat symbol is a unit vector in the direction of z.

 

[Merlin3189]

Thanks. I should have realized. It looked like a vector, but since they asked about the axial field, a vector seemed irrelevant. But I suppose it makes the equation mathematically right, with B indicated as a vector.

The more I look at it, I'm coming round to the idea that they're just asking, how far from the coil the field drops to half max. Then just double it because there is field on both sides of the coil.

Like you, I can't see any fitting. Not even parameters, just values of parameters to be substituted, * so that you can work out B for various z (or whatever method you use to find B(z) = (1/2 )B_max)

And the only one of them that looks at all interesting is R. The others affect the max field and the field at a distance equally.

Edit: * I shouldn't have said that. It may mislead you. Please ignore for the purposes of finding the FWHM.

 

[kuruman]

According to the my professor, the z hat symbol is a unit vector in the direction of z.

That is certainly the case. The B-field on the axis is an even function in z so it points in the $\hat z$ direction above and below the xy-plane. I assumed that the problem is asking to find the FWHM of the magnitude of the B-field on the z-axis, as the FWHM of a vector quantity is meaningless.

And the only one of them that looks at all interesting is R. The others affect the max field and the field at a distance equally.

In fact the scaling parameter $\zeta=z/R$ is the best way to describe the FWHM.

 

[mitchy16]

That is certainly the case. The B-field on the axis is an even function in z so it points in the $\hat z$ direction above and below the xy-plane. I assumed that the problem is asking to find the FWHM of the magnitude of the B-field on the z-axis, as the FWHM of a vector quantity is meaningless.

In fact the scaling parameter $\zeta=z/R$ is the best way to describe the FWHM.

Thank you for your help! I understand the graphing part and graphed it as you had stated but does that mean I don't have to use the formula they have provided?

 

[kuruman]

Thank you for your help! I understand the graphing part and graphed it as you had stated but does that mean I don't have to use the formula they have provided?

Not really. If you have find the FWHM in terms of z/R and you want it in meters then you need to multiply that FWHM by 0.105 m. The other variables/parameters do not affect the FWHM as @Merlin3189 has already remarked.

 

[mitchy16]

Not really. If you have find the FWHM in terms of z/R and you want it in meters the you need to multiply that FWHM by 0.105 m. The other variables/parameters do not affect the FWHM as @Merlin3189 has already remarked.

Thank you so much!