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(B) The direction of the magnetic field at every point on the axis of a current-carrying circular coil remains the same,although its magnitude varies.

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(B) The direction of the magnetic field at every point on the axis of a current-carrying circular coil remains the same,although its magnitude varies. Therefore,the magnetic field $B$ always remains positive along the axis.Consequently,graphs $(3)$ and $(4)$ are incorrect as they show negative values for the magnetic field.The magnitude of the magnetic field on the axis of a circular coil is given by:$B = \frac{\mu_{0} N i a^{2}}{2(a^{2} + x^{2})^{3/2}}$where $a$ is the radius of the coil,$N$ is the number of turns,and $i$ is the current.At the origin $(x = 0)$,the magnetic field is maximum:$B = \frac{\mu_{0} N i}{2a}$As $x \rightarrow \pm \infty$,the magnetic field $B \rightarrow 0$.The slope of the graph is given by the derivative:$\frac{dB}{dx} = -\frac{3 \mu_{0} N i a^{2} x}{2(a^{2} + x^{2})^{5/2}}$At $x = 0$,the slope $\frac{dB}{dx} = 0$. This means the tangent to the graph at $x = 0$ is parallel to the $x-$ axis,indicating a local maximum.Comparing this behavior with the given options,graph $(2)$ correctly represents the variation of the magnetic field along the $x-$ axis.

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