What do we call a 31 sided polygon ?

Magnetic field on the axis of a curent carrying ring of radius \(R\) at a distance \(x\) from the centre of the ring is given by

\(\frac { \mu _{ 0 }I }{ 2({ x }^{ 2 }+{ R }^{ 2 })^{ 3/2 } }\) ; where \(I\) is the current flowing through the ring.

The result can be derived either from

  • integration approach or

  • by deriving a result for \(n\) sided regular polygon and then taking the limit as \(n\rightarrow \infty \).

Finally, your job is to find the magnitude of the magnetic field on the axis of a \(31\) sided regular polygon of sidelength \(l\) carrying current \(I\) at a distance \(x\) from the centre of the polygon. If the magnitude of magnetic field comes out to be \(a\),

Give your final answer as \(a\times \ {10 }^{ 7 }\)

  • Details and Assumptions

  • \(l=0.1m\), \(x=0.1m\), \(I=1ampere\) .

  • \({ \mu }_{ o }\ =\ 4\pi \times \ { 10 }^{ -7 }\)
  • \(\sin { \frac { \pi }{ 31 } } \approx \ 0.1\) and so the other Trigonometric ratios can be calculated from it.
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