# Basic algebra problem for a magnetic field

After deriving the basic formula for a magnet field, $$\textbf{B} = \frac{\mu_0}{2\pi} \cdot \frac{Md}{\left(d^2 - l^2\right)^2}$$ where $$\textbf{B}$$ is the magnetic field strength, $$\mu_0$$ is the vacuum permeability of iron (0.25 H/m), $$M$$ is the magnetic dipole moment, $$d$$ is the distance of one magnet to another where the second's field is negligible, and $$l$$ is the supposed difference between the point charges of the magnet, North and South, inside the magnet. I'm trying to solve for $$l$$ however after performing another experiment I found this to be $$\approx 0.0215 \pm 0.0001$$ which when graphed is the only equation that follows a straight line (as in it is the only $$l$$ value that satisfies this correlation). I was hoping there was a method to solve for $$l$$ given that when it is taken as negligible and $$\textbf{B}$$ is plotted against $$1/d^3$$ the $$y$$-intercept is 0.7816, wherein you need to add 3.013. I feel as if there is an easier method then to rearrange to get a quartic equation (using wolfram alpha its solution is $$\approx 1$$ page using the Ferrari Method so...). I've also tried approaching it from the perspective of a contour integral with poles at $$\pm l$$ but it didn't end up working.

• Please be more explicit. Are you trying to identify a linearized functional form, such that you can plot the data and use information off the plot to identify the value of "l" (lower case L)? Are all values except "l" known? – David White Dec 13 '18 at 20:54
• $M$, $\mu_0$, $l$ are all constants while $\textbf{B}, d$ vary. I have plotted $\textbf{B}$ (y-axis) against $d$ to see how magnetic field strength changes with distance. When I ignore $l$ and take it as 0 I get an equation where it does not go through the origin, in fact it has a y intercept of 0.7816 but thats ignoring all the other constants. When I do the same thing but with the original equation, constants including $l$ taken as the measured value ($0.0215 \pm 0.0001\text{m})$ I get a curve that passed through the origin, so how can I find this $l$ without performing the 2nd experiment? – John Miller Dec 13 '18 at 21:04
• I take it that you need a linearized equation that you can plot. – David White Dec 13 '18 at 21:15
• Yes but I've found the equation for when it is a direct straightline passing through the origin and it only behaves like this when I substitute $l$ into the equation for $\textbf{B}$, when I take it as 0, as in there is no separation of charge, it does not make a straight line passing through the origin. Since I determined $l$ experimentally directly ($0.0215\pm 0.0001 \text{m}$) I was hoping to find it without using a second experiment, but this has become rather complex as I described before. – John Miller Dec 13 '18 at 21:19
• Is this a high school problem that you are working on? – David White Dec 13 '18 at 21:24