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Physics-

General

Easy

Question

A long insulated copper wire is closely wound as a spiral of N turns. The spiral has inner radius a and outer radius b. The spiral lies in the $X minus Y$ plane and a steady current $I$ flows through the wire. The Z-component of the magnetic field at the centre of the spiral is

900553

$\frac{μ_{0} N I}{2 (b - a)} \ln (\frac{b}{a})$
$\frac{μ_{0} N I}{2 (b - a)} \ln (\frac{b + a}{b - a})$
$\frac{μ_{0} N I}{2 b} \ln (\frac{b}{a})$
$\frac{μ_{0} N I}{2 b} \ln (\frac{b + a}{b - a})$

The correct answer is: $fraction numerator mu subscript 0 end subscript N I over denominator 2 left parenthesis b minus a right parenthesis end fraction ln invisible function application open parentheses fraction numerator b over denominator a end fraction close parentheses$

If we take a small strip of $d r$ at distance $r$ from centre, then number of turns in this strip would be,
$d N equals open parentheses fraction numerator N over denominator b minus a end fraction close parentheses d r$
Magnetic field due to this element at the centre of the coil will be
$d B equals fraction numerator mu subscript 0 end subscript open parentheses d N close parentheses I over denominator 2 r end fraction equals fraction numerator mu subscript 0 end subscript N I over denominator 2 open parentheses b minus a close parentheses end fraction fraction numerator d r over denominator r end fraction$
$B equals not stretchy integral subscript r equals a end subscript superscript r equals b end superscript d B equals fraction numerator mu subscript 0 end subscript N I over denominator 2 open parentheses b minus a close parentheses end fraction ln invisible function application open parentheses fraction numerator b over denominator a end fraction close parentheses$

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