Can I Just Change the Coil in Circuits to Find the Relationship Between B and r?

[PhysicStud01]

Homework Statement

Homework Equations

none

The Attempt at a Solution

I want to find the relationship between B and r. So, i have to change r.
Do I just change the coil completely with a new one? I believe that this could greatly affect the set up and I don;t think it is practical. OR is there a basic thing that I am missing?

 

[kuruman]

It is not at all practical to change $r$, I agree. You are missing that the coil separation $d$ must be part of the expression for the B-field at the midpoint. The expression that you are given is valid for only for a fixed ratio $\beta = d/r$. Thus, if you want to test the relationship, you must vary both $r$ and $d$ in a way that keeps the ratio constant. In other words, the relationship you are given appears to have $r$ as an independent variable, but it is not independent. The numerical factor 0.72 implicitly contains the ratio $\beta$. I strongly recommend that you derive a general expression for the B-field at the midpoint when the separation is $d$ and the radius is $r$. It will help you see what's going on and see where the factor 0.72 is coming from.

One thing you could do experimentally is to test the given expression is to measure the B-field at the midpoint as a function of coil separation $d$ and find for what value the given relationship is valid. However, I am not sure that this is what the exercise expects you to do.

 

[PhysicStud01]

It is not at all practical to change $r$, I agree. You are missing that the coil separation $d$ must be part of the expression for the B-field at the midpoint. The expression that you are given is valid for only for a fixed ratio $\beta = d/r$. Thus, if you want to test the relationship, you must vary both $r$ and $d$ in a way that keeps the ratio constant. In other words, the relationship you are given appears to have $r$ as an independent variable, but it is not independent. The numerical factor 0.72 implicitly contains the ratio $\beta$. I strongly recommend that you derive a general expression for the B-field at the midpoint when the separation is $d$ and the radius is $r$. It will help you see what's going on and see where the factor 0.72 is coming from.

One thing you could do experimentally is to test the given expression is to measure the B-field at the midpoint as a function of coil separation $d$ and find for what value the given relationship is valid. However, I am not sure that this is what the exercise expects you to do.

we need to design an experiment to test the relationship between B and r. so, i think that we need to change r. But is there any way that does not disturb the set-up or should i just change the coil entirely.
i don't think that we need to go into the ratio you have given in this question

 

[kuruman]

My point is that the formula is valid only for a given value of the ratio $\beta=d/r$. If you keep the coil separation the same and change only the radius, you are not testing the relationship that was given to you with $r$ as an independent variable. Stated differently, the correct expression for the B-field at the midpoint is $$B=\frac{f(\beta)\mu_0NI}{r}$$The coils that are given to you are in the Helmholtz configuration ($\beta = 1$) in which case $f(1)=(4/5)^{3/2}=0.72$. If you change $r$ only, you also change $\beta$ in which case $f(\beta)$ has a different value from $0.72$.

I will repeat my strong recommendation: if you want to test the expression that is given to you, you need to at least understand where it comes from and how it is put together.