# Relation Between the Magnetic Field Strength and Distance

## Main Question or Discussion Point

Hey guys, I have recently performed a lab where we are to find the equation relating the magnetic field strength to a distance from a dipole in both the transverse and longitudinal direction. We did this using a Hall Effect Probe and a magnetic dipole.

What I found after plotting a Log Voltage (V) Vs. Log Distance (m) is a slope of -3, which represents an exponential power.

After doing some research, I found that the magnetic field strength will be inversely proportional to that of the distance, with a ratio of 1/x^3.

The relation between the slope and the cubed value makes me believe that there is a relationship and that I am on the right track. However, I am not exactly sure what to do from here, and I am not entirely sure as to how the relationship is formed.

Currently, I am tempted to use the formula:

Bn = m/x^2(1-L/2x)^2 and Bs = -m/x^2(1-L/2x)^2 to find the vector sum so that I can use the electric field of from some distance x away, but I'm not sure what that value will even provide me.

If you could provide any information regarding the this relationship, or knowledge relative to this subject, I will be more than happy for your help.

EDIT: I am also wondering what a plot of the Voltage Vs Distance^-3 (x^-3) on a linear graph would produce... If Y = mx + b is found from this graph, will this lead me to the formula as well? I'm still confused.

Aaaand I realized that I did not post this to the homework sub-forum. Sorry everybody.

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Simon Bridge
Homework Helper
You are doing an experiment - which means you have to use whatever you data tells you.

You have found that the quantity plotted on one axis is related to the quantity plotted on another axis, and you have that relationship. Your next step is to relate what you plotted with magnetic field strength. You should know what this relationship is already - i.e. before you plotted the graph. Else: why did you plot the graph?

I know - you were following instructions (I'm guessing) before you understood what they were for right?

So go back - why did they tell you to plot those particular values?
What is the relationship between the hall voltage and the strength of the magnetic field?

1 person
My present data suggests that as the distance increases, the hall voltage decreases, showing an inverse relationship between these two characteristics. The slope of -3 when plotting the graph shows that the relationship can be represented as 1/x^3. Which, as far as my guess goes, means that the hall voltage will equal 1/x^3 for some distance x and thus the strength of the magnetic field will decrease as x increases.

As the dipole moves to distances that are "far away" , the magnetic field will appear as 1/x^3.

So by plotting a Voltage vs 1/x^3 on a linear graph, a relationship will be shown between the voltage and the magnetic field strength due to distance? And thus a relationship between the measured Hall voltage and the magnetic field strength can be seen as well?

Simon Bridge
Homework Helper
My present data suggests that as the distance increases, the hall voltage decreases
That is the relationship between the hall voltage and the distance to the magnet.
But what is the relationship between the hall voltage and the magnetic field strength?

Do you know how your detector works?

1 person
Based on the article I've read here: http://www.explainthatstuff.com/hall-effect-sensors.html

The hall voltage will measure the potential difference across a material, as one end of the dipole will experience a varying amount of electrons than the other. When there is a large magnetic field, the Hall voltage will also be large as well. Similarly, when the magnetic field is small, the Hall voltage will decrease.

Which, based on my results, will signify that there will be a direct relationship between the distance and the magnetic field strength. As one decreases, the other will decrease as well. The rate for these two variables is unknown at the moment, but because the hall voltage and magnetic field strength are directly proportional, this means that when these points are subject to a ratio, there will be a constant that is produced as well.

EDIT: So if I want to view the relationship between the Hall voltage and magnetic field strength, I can calculate the field strength using Bn = m/x^2(1-L/2x)^2 and Bs = -m/x^2(1-L/2x)^2, find their vector sum, and plot it on a graph Vs the Hall Voltage? Not exactly sure how that would relate into an equation with my 1/x^3 slope found previously though.

Simon Bridge
Homework Helper
Based on the article I've read here: http://www.explainthatstuff.com/hall-effect-sensors.html

The hall voltage will measure the potential difference across a material, as one end of the dipole will experience a varying amount of electrons than the other. When there is a large magnetic field, the Hall voltage will also be large as well. Similarly, when the magnetic field is small, the Hall voltage will decrease.
... I see you realized that is not what I was asking you for... after thinking about it you did an edit:
EDIT: So if I want to view the relationship between the Hall voltage and magnetic field strength, I can calculate the field strength using B_n = m/x^2(1-L/2x)^2 and Bs = -m/x^2(1-L/2x)^2, find their vector sum, and plot it on a graph Vs the Hall Voltage? Not exactly sure how that would relate into an equation with my 1/x^3 slope found previously though.
You've figured out that when you are asked for the scientific relationship between two things you can measure, your answer should be some math of some sort that describes that relationship, an equation is best. Math is a language too - use it like one.

You already used the graph to get a relationship (equation) between the hall voltage and the distance ... something like ##V=mx+c## - it's linear, right?

What you need now is the equation relating the hall voltage to the magnetic field.
Something like ##V=f(\vec{B})## ... i.e. the voltage is some function of the magnetic field.

Then you can use algebra to relate the two equations ... from above you get: ##f(\vec{B})=mx+c## ... see what I mean? But you need to know what ##f(\vec{B})## is, in order to do this.

Ideally you want the final equation to depend on m, the slope of your graph, alone. (Does your experimental line go through the origin?)

A slope of m in your V vs x graph probably means that ##B=k/x^m## (which you found out by looking it up, right? - but that may not apply here so: ) ... basically you are checking this with the algebra.

You've already discovered a complication in that the orientation of the sensor matters :)
There should be a theory section with the instructions you used to complete the lab. That normally has all the equations, with a short explanation of what they do, with it. Some courses cover this material in class though - so if you miss a class, you will have a hard time.

It may be you only need to say that "B has an inverse-m relationship with distance". But it is more likely, and much better if, you need to figure out what the constant of proportionality, k, is equal too as well.

1 person
Regarding the aspect of the sensor orientation, it did mention that the Hall Effect Probe needed to be rotated 90 degrees when switching the dipole from the transverse axis to the longitudinal axis.

After plotting a Voltage Vs Distance (x^-3) graph, the resulting slope will give me the constant 'k' in the expression. V = k(x^-3). However, because the Hall Voltage and the electric field strength are directly proportional, the rate at which the voltage changes will have an equivalent effect on the field strength... giving us:

B = k(x^m)

Which also coincides with the equation you have given as well. The data points passes through the origin of the graph as well. Therefore, the log-log plot has given us the power relationship needed for the exponent "m", and a graph of the voltage and the distance to the power of m will give us some value (with error) that will represent the proportionality constant "k".

Thus, we are able to conclude that the equation given of the magnetic field strength with respect to a distance x is given by the formula found above?

EDIT: Though, my results have shown that the slope of the linear graph will have units of V(m^3). And when this is multiplied by the distance (x^-3) which will yield a unit of m^-3, the result will be in volts. Meaning perhaps the magnetic field strength cannot be substituted in place of V in the above equation even if they are proportional. Which now leads to the final question. How can I show that the equation V = k(x^m) is directly proportional to the magnetic field strength?

SECOND EDIT: From the slope that I obtained as my k value, I can use the error to test the boundaries of the maximum and minimum values for V = k(x^-3). Then, I could use the formulas: Bn = m/x^2(1-L/2x)^2 and Bs = -m/x^2(1-L/2x)^2 to determine the magnetic field strength at some distance x after finding its vector sum, and use the same distance for x in (x^-3).

If the result of the field strength is within the maximum and minimum range of the voltage for multiple distances, then I would be able to conclude that the voltage and the magnetic field strength are, in fact, directly proportional.

So far, my results using this method have proven to relate. Success.

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Simon Bridge