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Physics-

General

Easy

Question

An infinitely long wire carrying current $i$ is along $Y$ -axis such that its one end is at point ( $0 comma blank b right parenthesis$ while the wire extends upto $infinity$ . The magnitude of magnetic field strength at point $P blank left parenthesis a comma blank 0 right parenthesis$ is

$\frac{μ_{0} i}{4 π a} (1 + \frac{b}{\sqrt{a^{2} + b^{2}}})$
$\frac{μ_{0} i}{4 π a} (1 - \frac{b}{\sqrt{a^{2} + b^{2}}})$
$\frac{μ_{0} i}{4 π a} (1 - \frac{a}{\sqrt{a^{2} + b^{2}}})$
$\frac{μ_{0} i}{4 π a} (\frac{b}{\sqrt{a^{2} + b^{2}}})$

The correct answer is: $fraction numerator mu subscript 0 end subscript i over denominator 4 pi a end fraction open parentheses 1 minus fraction numerator b over denominator square root of a to the power of 2 end exponent plus b to the power of 2 end exponent end root end fraction close parentheses$

As shown figure take an element $d l blank a t blank C blank o f blank w i r e comma blank w h e r e blank O C equals l. blank L e t blank P C equals r blank a n d blank angle O P C equals blank ϕ$ .
According to BiotSavarts law; magnitude of magnetic field induction at $P$ due to current element at $C$ is

$d B equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i d l sin invisible function application theta over denominator r to the power of 2 end exponent end fraction$ (i)
Here, $theta blank equals 90 degree plus ϕ semicolon r equals a sec invisible function application ϕ$
And $l blank equals a tan invisible function application ϕ semicolon d l equals a sec to the power of 2 end exponent invisible function application ϕ blank d blank ϕ$
$therefore blank d B equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator i left parenthesis sec to the power of 2 end exponent invisible function application ϕ blank d blank ϕ right parenthesis sin invisible function application left parenthesis 90 degree plus ϕ right parenthesis over denominator a to the power of 2 end exponent sec to the power of 2 end exponent invisible function application ϕ end fraction$
$blank equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 1 over denominator a end fraction cos invisible function application ϕ blank d blank ϕ$
Total magnetic field induction at $P$ is
$B blank equals blank not stretchy integral subscript ϕ subscript 1 end subscript end subscript 90 degree fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 1 over denominator a end fraction cos invisible function application ϕ blank d blank ϕ equals blank fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 1 over denominator a end fraction blank left parenthesis sin invisible function application ϕ right parenthesis blank subscript ϕ end subscript subscript 1 end subscript superscript 90 end superscript degree$
$equals blank fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 1 over denominator a end fraction left parenthesis 1 minus sin invisible function application ϕ subscript 1 end subscript right parenthesis equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 1 over denominator a end fraction open parentheses 1 minus fraction numerator b over denominator square root of a to the power of 2 end exponent plus b to the power of 2 end exponent end root end fraction close parentheses$

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