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I have a cylindrical flat permanent magnet with the radius $r$ and height $2h$ (in the origin of the coordinate system) made of the material with a constant magnetization $\vec M = M \vec e_z$. And I should find the asymptotic behavior for the magnetic induction $\vec B$ if $z \to \infty$.

$\vec B = \frac{k \vec e_z}{(a^2 + z^2)^{\frac{3}{2}}}$

  • The only thing that came up to my mind is to calculate the limit $z \to \infty$ which would be zero and I think it's not what I should to. The problem is I don't actually understand what I should do to find the asymptotic behavior because I haven't encounter problem like this yet. Any hint how to start please?

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The answer should be $\vec B = \frac {k \vec e_z}{z^n}$, and I don't understand why is there that $n$.

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2 Answers 2

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Hint: the bigger $z$ becomes the less significant the value of $a$ is.

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As $z \to \infty$, the $a$ in $a^2 + z^2$ becomes negligible and thus we get, $$\vec B = \frac{k \vec e_z}{(a^2 + z^2)^{\frac{3}{2}}} = \frac{k \vec e_z}{(z^2)^{\frac{3}{2}}}= \frac{k \vec e_z}{z^3} $$

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  • $\begingroup$ The answer is actually $\vec B = \frac {k \vec e_z}{z^n}$ and I don't understand why. $\endgroup$
    – user156350
    Sep 25, 2017 at 21:38
  • $\begingroup$ the formula which you gave is for the field due to a current carrying coil along its axis. Under the above simplifications, it reduces to the field of a magnetic moment along it's axis. $\endgroup$ Sep 26, 2017 at 3:45

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