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(A) For a current-carrying circular coil,the magnetic field lines originate from the coil and form closed loops. By Gauss's law for magnetism,the net

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$\phi_{i} = -\phi_{0}$

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(A) For a current-carrying circular coil,the magnetic field lines originate from the coil and form closed loops. By Gauss's law for magnetism,the net magnetic flux through any closed surface is zero,i.e.,$\oint \vec{B} \cdot d\vec{A} = 0$. Consider the infinite plane containing the coil. The magnetic flux passing through the area of the coil $(\phi_{0})$ is in one direction (e.g.,upward). The magnetic flux passing through the rest of the infinite plane $(\phi_{i})$ must be in the opposite direction (downward) to ensure that the total flux through the entire infinite plane is zero. Thus,$\phi_{0} + \phi_{i} = 0$,which implies $\phi_{i} = -\phi_{0}$.

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