Magnetic field strength dependent on distance from source

[babemagnetics]

I am trying to find an equation that tells the strength of a magnetic field a given distance away from the source. It would be very helpful if all terms are defined, since the internet is notorious for not saying what variables mean.

Gracias

 

[vanhees71]

In leading order the magnetostatic field far away from the sources (permanent magnets or currents) goes like 1/r^3, where r is the distance from the source. The multipole expansion for magnetic fields always starts with the dipole term since there are no magnetic monopoles (at least nobody could find any yet). I think, the corresponding Wikipedia article is a quite good source for first orientation:

http://en.wikipedia.org/wiki/Dipole#Vector_form

 

[babemagnetics]

In leading order the magnetostatic field far away from the sources (permanent magnets or currents) goes like 1/r^3, where r is the distance from the source. The multipole expansion for magnetic fields always starts with the dipole term since there are no magnetic monopoles (at least nobody could find any yet). I think, the corresponding Wikipedia article is a quite good source for first orientation:

http://en.wikipedia.org/wiki/Dipole#Vector_form

First of all, thank you. Second, on the Wikipedia page, I don't see the equation you provided; is the strength of a magnetic field really as simple as just multiplying the magnetic field at the source by 1 divided by the cube of the distance you are from the magnet? Also, what are the units that 1/r^3 is made to work with please (Tesla? Gauss?)?

Gracias

 

[Philip Wood]

It depends what you mean by 'source'. Replies so far assume you mean a current loop. In that case, for a plane loop carrying current I around the edge of a 'hole' of area A, the field on the axis of the loop, and a long way, x, from the loop is given by
B = \frac{\mu_{0} I A}{2\pi x^3}

IA is called the magnetic moment, M, of the loop.

B comes out in tesla, if other quantities are entered in SI units.

Formulae for the field at points off-axis, a long way from the loop can also be given. But if you're closer to to the loop than, say, ten times its radius (or side length or other characteristic dimension) these formulae become increasingly inaccurate. More accurate ones are complicated and depend on the shape of the loop, not just its area.

Another approach is to regard a current element (a short length of conductor carrying a current) as a source. There is a famous formula for the field due to a current element, usually called the Biot-Savart Rule. It's an inverse square law, with the complication of an angular dependence. It's in all e-m texts. The field at a point in the vicinity of a circuit (that is a current loop of any shape) can be found by integrating up the fields at that point from all the current elements in the circuit, remembering that the fields are vectors.

 

[babemagnetics]

So the 1/r^3 equation is no good? That guy has a PhD. If 1/r^3 is correct, then would someone please tell me whether it's in Gauss or Tesla? Also, how far is "magnetostatic field far away from the sources (permanent magnets or currents)"? And I am assuming r is in meters?

Gracias once again

 

[vanhees71]

Philip Wood's reply is correct. It's the dipole approximation for a current loop, and he's given the magnitude of the magnetic field (formerly called magnetic induction) in SI units. So, as he said, his result gives you the magnetic field in Tesla, if you plug in the current in Ampere, the length in meter, and the area in \mathrm{m}^2.

 

[Philip Wood]

No, what vanHees77 said is correct (at large distances from the current loop). The formula I gave was also, please note, an inverse cube law - for for the simple special case of the field along the axis of the loop (that is along a line normal to the plane of the loop, passing through the centre of the loop. It included the 'constants', because I thought you wanted to know these. A more general formula for the magnitude of the field at a distance along way, r, from the centre of the dipole, and along a 'radius' at angle \theta to the axis is

B = \frac{\mu_{0}AI \sqrt{1 + cos^2(\theta)}}{4\pi r^3}

If you need to know the direction of the field at (r, \theta), you'll need a vector equation or scalar equations for both the radial and tangential field components.

All these equations give [B in tesla, as long as what you put into the formulae is SI.

Just to clear up a possible confusion generated by my last post... I said that the Biot-Savart law gave an inverse square law for the field due to a current element. What I might have added, is that if you apply it to a current loop, adding up the contributions from all the elements into which you split the loop, you get just exactly the inverse cube formula I've just quoted (at points a long way from the loop).

 

[babemagnetics]

Using 1/r^3, do you multiply 1/r^3 by the magnetic field source strength, or do you subtract 1/r^3 from the magnetic field source strength?

Gracias

 

[Born2bwire]

You multiply. The field falls off as 1/r^3. This is, as already stated by the others, a rough approximation since we have no knowledge of what your magnet is like. But it is generally a good guess that most magnetic sources are primarily dipoles.

 

[babemagnetics]

Gracias sir

 

[downtownjapan]

Can anyone add any information on calculating how the field changes when the distance is very, very small? For example 0.000075 meters? I am assuming the 1/r^3 formula does not apply at this distance.
I have a coil 0.00004 meters in diameter producing a magnetic field of 31.69 tesla. Can anyone suggest how I can measure the field strength at a point 0.000075 meters away from the center of the coil?
(I hope it is okay to post a question here and not an answer. I am trying to calculate the same type of thing as the original poster. Apologies if this is not the correct thing to do!)

 

[Philip Wood]

That's a VERY small coil and a VERY large field. Is this a real set-up or a theoretical problem?

Do you mean measure the field strength, or do you mean predict what it will be?

If the coil is a circular loop then it's easy to predict the flux density at any point on its axis. If you can't find the relevant formula, please ask. For off-axis points, it's very much harder!

 

[Philip Wood]

Downtownjapan. Have you received the information you wanted?

 

[mfb]

Looks problematic as real setup. As a current loop and with 31T field strength in the center, this would require ~1000A.
Assuming ~(40µm)^2 cross-section for the current loop (note that this does not fit within the described diameter), this corresponds to a current density of 600kA/mm^2 or 6*10^7 A/cm^2. I found values up to ~10^7A/cm^2 for superconductors - but those are at zero magnetic field!

Bigger current loops have a better scaling - the cross-section increases with diameter squared, while the distance term drops with diameter.

 

[tris_d]

It depends what you mean by 'source'. Replies so far assume you mean a current loop. In that case, for a plane loop carrying current I around the edge of a 'hole' of area A, the field on the axis of the loop, and a long way, x, from the loop is given by
B = \frac{\mu_{0} I A}{2\pi x^3}

Can you point some reference about that equation?

 

[jtbell]

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html#c3

Take the limit as z >> R.

 

[tris_d]

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html#c3

Take the limit as z >> R.

B = \frac{\mu_{0} I A}{2\pi x^3}

I don't see this equation was give explicitly. What is it supposed to be describing:

a.) Field at Center of Current Loop

b.) Field on Axis of Current Loop

c.) Something elseI could also not find it anywhere on Wikipedia, but I found something which confused me even more:

http://en.wikipedia.org/wiki/Biot–Savart_law

Electric currents (along closed curve):

1.)
\mathbf{B} = \frac{\mu_0}{4\pi} \int_C \frac{I d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3}

- where r is the full displacement vector from the wire element to the point at which the field is being computed and r̂ is the unit vector of r. Using this the equation can be equivalently written:

2.)
\mathbf{B} = \frac{\mu_0}{4\pi}\int_C \frac{I d\mathbf{l} \times \mathbf{\hat r}}{|\mathbf{r}|^2}
I think the first equation can not be without unit vector, and that it is CIRCLE integral and should be written like this:

1.)
\mathbf{B} = \frac{\mu_0}{4\pi} \oint_C \frac{I d\mathbf{l} \times \mathbf{\hat r}}{|\mathbf{r}|^3}

where top 'r' is unit vector which they missed to denote, and the second equation is LINE integral and should be written like this:

2.)
\mathbf{B} = \frac{\mu_0}{4\pi}\int_L \frac{I d\mathbf{l} \times \mathbf{\hat r}}{|\mathbf{r}|^2}Now, if I got that right, the question is why would circle integral be 1/r^3 instead of 1/r^2?

 

[mrv11]

I have a practical question to add...

Using a calculator found here: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html#c3

I can calculate the magnetic field within a given solenoid. To verify this I have a measured current flowing through the wire and a compass. I align the solenoid at a right angle to the Earth's magnetic field and bring the solenoid closer to the compass until it is deflected 45 degrees.

Therefore using vector analysis, I know the field produced by my solenoid at this distance to be equal to the field of the Earth at my location (roughly 4.0 x 10^-5 Tesla).

To verify the calculated magnetic field strength in my solenoid do I use the a re-arranged version of
B=(μ₀NIA)/(2∏x³) ?
N= number of turns

I cannot imagine B_solenoid=B_{at compass}r³ would work... The dimensional analysis would leave me with improper units!

Thank you in advance!

 

[jtbell]

See here for the magnetic field along the axis of a finite solenoid:

http://www.phys.uri.edu/~gerhard/PHY204/tsl215.pdf

This assumes the solenoid has no iron or other magnetic material inside it.

 

[mrv11]

Thank you all for the help on this little project!

I have to apologize about the delay for my thanks. School became overwhelming so I had to put this on hold. I reviewed the info you all provided, along with a great text a professor lent me. While not one of those individual sources described exactly my situation, I was able to synthesize it all into a decent model to describe my data.

Thanks again!

 

[downtownjapan]

Hi everyone,

I am trying to use the 1/r^3 formula to calculate the decrease in magnetic field strength over distance, but I must be doing something very, very wrong following this formula because the numbers I get don't make sense to me. I have clearly made a mistake applying the formula and I am wondering if someone could tell me where I have gone wrong.

I am trying to calculate how a field of 0.005 Tesla will decrease at a distance of 0.0005 meters away.

I have done the following calculation
0.005T/0.0005^3 meters, which gives me the number
0.005/0.0005^3 = 40,000,000

Obviously I have gone horribly wrong, and I have a feeling I am making a very basic mistake somewhere with either the calculations and/or the understanding of the formula. If any kind-hearted charitable soul out there wants to tell me how this poor fool has gone wrong, I would be very grateful!

 

[Philip Wood]

I am trying to calculate how a field of 0.005 Tesla will decrease at a distance of 0.0005 meters away.

Away from where? You need to sort out from where your distances are measured… We need to know the set-up.

 

[mfb]

Hi everyone,

I am trying to use the 1/r^3 formula to calculate the decrease in magnetic field strength over distance, but I must be doing something very, very wrong following this formula because the numbers I get don't make sense to me. I have clearly made a mistake applying the formula and I am wondering if someone could tell me where I have gone wrong.

I am trying to calculate how a field of 0.005 Tesla will decrease at a distance of 0.0005 meters away.

I have done the following calculation
0.005T/0.0005^3 meters, which gives me the number
0.005/0.0005^3 = 40,000,000

Obviously I have gone horribly wrong, and I have a feeling I am making a very basic mistake somewhere with either the calculations and/or the understanding of the formula. If any kind-hearted charitable soul out there wants to tell me how this poor fool has gone wrong, I would be very grateful!

A simple way to see why your answer is wrong: work with units. Your calculation gives 40000000 T/m³ - this has the wrong units for a magnetic field strength. And the numeric value depends on the length units you use.

Those 40000000T would be the magnetic field at this distance if you had a point-like source that has a field of 0.005 T in a distance of 1m. Such a source is completely unrealistic.

By the way: please start a new thread if other threads are several months old.

 

[Angry Bear]

The original answer 1/r^3 is unitless because it's a ratio.

You can simply use ratios. If B1 = (a bunch of crazy symbols)/x^3 and B2 = (same bunch of crazy symbols)/y^3 then B1/B2 = y^3/x^3. So if you know x and y and you know B1 then you can find B2 with any units of x and y as long as they're the same units and any units of B1 and B2 as long as they are the same units, call them strength or magnetic flux.

All approximations and conjecture aside, if you have a magnet of 1lb pull strength at 1 inch and you increase the distance to 2 inches, the pull strength would be 1/8th.

You're welcome.

 

[anonymouskkkk]

Hey everyone,
i have understood the calculation to get the magnetic field strength at a large distance.
But is there any equation that works for very short distances from magnet?
thanks,

 

[Philip Wood]

Magnets are not simple things. Their material is magnetised, but the magnetisation will vary somewhat in both magnitude and direction from point to point in the magnet, being pretty uniform at the centre but 'splaying out' at the ends. Therefore I think it unlikely that there'll be any simple equation that gives the field strength outside the magnet, except at a long distance away, when the magnet behaves as a dipole, with an inverse cube law field.

I think your best bet is an experimental investigation using a Hall probe.

 

[mfb]

But is there any equation that works for very short distances from magnet?

Nothing easy, you typically need a finite-element simulation.

This thread is from 2011, please open a new thread.