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A thin insulated wire forms a plane spir...

A thin insulated wire forms a plane spiral of `N=100` turns carrying a current `i=8mA`. The inner and outer radii are equal to `a=5 cm and b=10 cm`. Find the magnetic induction at the centre of the spiral

Text Solution

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Let n be the number of turns per unit length along the radii
of the spiral. Consider a ring of radii x and x+dx.

Number of turns in the ring =`n dx`
`:.` Magnetic fild at the centre due to this ring =`(mu_0(n dx)i)/(2x)`
`:. B("total field") =int _a^b(mu_0nidx)/(2x)=1/2mu_0ni ln (b/a)`
But `n=N/(b-a)`
`B=(mu_0Ni)/(2(b-a))ln (b/a)=(4pixx10^-7xx100xx8xx10^-3)/(2(10-5))xx10^-2) ln(10/5)`
`=6.9xx10^-6T=6.9muT`
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