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Question

At what value of x is the magnitude of $$ \overrightarrow {B} $$ is maximum ? 


A
x=0
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B
x=2a
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C
x=±a/2
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D
x=±a
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Solution

The correct option is D $$ x = \pm a $$
At a position on the x-axis :
$$ B_{net} = 2 \frac { \mu_0 I}{ 2 \pi  r }  cos \theta  = \frac { \mu_0 I}{ \pi \sqrt { x^2 + a^2 } } \frac {x}{ \sqrt { x^2 +a^2 } } $$
$$ B_{net} = \frac { \mu_0 Ix }{ \pi ( x^2 +a^2 ) } $$
its graph is shown below.
The magnetic field is maximum when :
$$ \frac {dB}{dx} = 0 = \frac { C}{ x^2 +a^2 } - \frac { 2Cx^2 }{ (x^2 +a^2 )^2} $$
$$ ( x^2 +a^2 ) - 2x^2 \Rightarrow x  = \pm a $$ .

1620682_1747574_ans_3b11a6eaa9264b078712d7656de66218.png

Physics
NCERT
Standard XII

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