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

  1. 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    
  2. 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 plus a over denominator b minus a end fraction close parentheses    
  3. fraction numerator mu subscript 0 end subscript N I over denominator 2 b end fraction ln invisible function application open parentheses fraction numerator b over denominator a end fraction close parentheses    
  4. fraction numerator mu subscript 0 end subscript N I over denominator 2 b end fraction ln invisible function application open parentheses fraction numerator b plus a over denominator b minus a end fraction close parentheses    

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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