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A homework question has made me doubt my understanding of the application of the Biot-Savart law.

Question: The magnetic field 43.0 cm away from a long, straight wire carrying current 5.00 A is measured to be 8.00 μT. At what distance is it 0.800 μT?

I thought this was a simple rearrange and substitute problem, so I assumed the wire was significantly longer than the .43 m radial distance and then immediately bumped into my first problem,

$$\vec B=\frac{\mu_oI}{2\pi r}=\frac{2\times 10^{-7}(5)}{0.43}\not=8.00 μT$$


Progress

I drew the following representation (note actual situation is not symmetrical about the vertical axis) enter image description here

The objective is to determine the radial distance $r$ from the wire that produces a magnetic field $\vec B$ that is $0.8\mu T$ or one tenth of $\vec B_0$.

I abandoned the assumption that the wire could be considered infinitely long in comparison the the radial distance and returned to:

$$\vec B_o=\frac{\mu_oI}{4\pi a}\cdot (\sin\theta_3 - \sin\theta_4)$$ Substituting values from the question the quantity for $(\sin\theta_3 - \sin\theta_4)$ is found as $6.88$ At some greater radial distance $r$: $$\vec B=\frac{\vec B_o}{10}=\frac{\mu_oI}{4\pi r}\cdot (\sin\theta_1 - \sin\theta_2)$$ Now I have two unknown quantities ($r$ and $\sin\theta_1 - \sin\theta_2$) but only one equation $|\vec B|=0.8\mu T$.


Is there a link between $(\sin\theta_3 - \sin\theta_4)$ and $(\sin\theta_1 - \sin\theta_2)$ That will allow me to reduce this to one unknown one equation?

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    $\begingroup$ I get B(r=0.43m) = 2.3 μT. Seems like the problem is incorrectly constructed. It's overconstrained anyways - there's no need to give the distance, the current, and the field strength - two of those determines the third. $\endgroup$ – Brionius Dec 23 '14 at 20:43
  • $\begingroup$ Thank you, yes I get the same. I'm sure though that the question is legit, it came from Physics Vol2. Serway, Jewett, Wilson. $\endgroup$ – Nic Dec 23 '14 at 20:54
  • $\begingroup$ Assuming the question is not wrong, don't give up. You tried to do away with the assumption that it is infinitely long wire. That seems like the right path $\endgroup$ – Cheeku Dec 24 '14 at 12:34
  • $\begingroup$ I think that there is something wrong with the question since no good question gives you both the means to find something AND actually give you that something!There is no logic in it! Now,if the question in your book said that it is a long wire,then it must imply that is has infinite length,so the first equation that you used must be correct. $\endgroup$ – TheQuantumMan Mar 24 '15 at 16:56
  • $\begingroup$ The question is wrong. It is not possible for difference between two sine values to be greater than 2. $\endgroup$ – ShankRam Dec 6 '15 at 16:51
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Well it may be that it was not free space where magnetic field was measured. Usually materials having value of myu smaller than that of free space will show less strength of magnetic field.

The typical values of myu for above mentioned materials is 0.999 times that of free space. No lower value of myu has been found except superconductors which have almost zero myu.

Hence question is incorrectly framed.

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