Question

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

A
x=0
B
x=2a
C
x=±a/2
D
x=±a

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 :$$\frac {dB}{dx} = 0 = \frac { C}{ x^2 +a^2 } - \frac { 2Cx^2 }{ (x^2 +a^2 )^2}$$$$( x^2 +a^2 ) - 2x^2 \Rightarrow x = \pm a$$ .PhysicsNCERTStandard XII

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