Is This Calculation of Magnetic Field at the Center of a Solenoid Correct?

[vaxopy]

B = uNI/2L * (L/2/(\sqrt{((L/2)^2)+R^2)} - (-L/2/(\sqrt{((-L/2))^2+R^2))}

I get.. B = uNI/2R (for those wondering, this is the equation to find the magnetic field when a Hall probe is placed in the center of a solenoid)

they ask me to figure out the % diffeerence between the field at the center of the solenoid compared to a solenoid that infinitely long (where B = uNI)..

I get L = 28cm, R = 2.69cm, I = 1.25A

i get 520% when using my simplified formula >_<
is this right?

and i don't even use the current!

 

[HallsofIvy]

It's hard to tell- you are clearly missing at least one parenthesis. Is the initial
(uNI/2L) multiplied by BOTH fractions (in which case they clearly add) or is it multiplying only the first fraction (which is what you have written- and is much harder).

 

[thepatientmental]

Sure, I'd be happy to help simplify this equation for you! First, let's break down the given equation into smaller parts:

- u: This represents the permeability of the medium, which is a constant value.
- N: This represents the number of turns in the solenoid coil.
- I: This represents the current flowing through the solenoid.
- L: This represents the length of the solenoid.
- R: This represents the radius of the solenoid.

Now, let's take a closer look at the first part of the equation: uNI/2L. We can simplify this by dividing both the numerator and denominator by 2, giving us uNI/L. This is because dividing by 2 in both the numerator and denominator is the same as dividing by 2L.

Next, let's look at the second part of the equation: (L/2/(\sqrt{((L/2)^2)+R^2)} - (-L/2/(\sqrt{((-L/2))^2+R^2)). This part can be simplified by first rewriting it as (L/2)/\sqrt{((L/2)^2)+R^2} - (-L/2)/\sqrt{((-L/2))^2+R^2}. This is because the division sign can be rewritten as a fraction. Next, we can simplify the fractions by multiplying both the numerator and denominator by 2, giving us L/\sqrt{(L^2)+4R^2} - (-L/\sqrt{(L^2)+4R^2}. Simplifying further, we get L/\sqrt{(L^2)+4R^2} + L/\sqrt{(L^2)+4R^2}. Finally, we can combine these two terms by adding them together, giving us 2L/\sqrt{(L^2)+4R^2}.

Now, let's put everything back together: B = uNI/L * 2L/\sqrt{(L^2)+4R^2}. We can further simplify this by canceling out the Ls, leaving us with B = uNI/\sqrt{(L^2)+4R^2}. This is the simplified form of the original equation.

To answer the question about the difference between the magnetic field at the center of a finite solenoid compared to an infinitely long solenoid, we can use this simplified