Consider a circular coil of wire carrying constant current I, forming a magnetic dipole. The magnetic flux through an infinite plane that contains the circular coil and excluding the circular coil area is given by $$\phi_i$$. The magnetic flux through the area of the circular coil area is given by $$\phi_0$$. Which of the following option is correct?

Total magnetic flux through any closed Gaussian surface: $$\oint \vec{B} \cdot d\vec{A} = 0$$

Let a closed surface encompass the entire upper hemisphere bounded by the infinite plane:

$$\phi_{\text{total}} = \phi_{\text{coil\_area}} + \phi_{\text{remaining\_plane}} + \phi_{\text{hemisphere\_at\_infinity}} = 0$$

Since magnetic field falls off as $$B \propto \frac{1}{r^3}$$ for a dipole: $$\phi_{\text{hemisphere\_at\_infinity}} = 0$$

Flux contribution from the two regions of the infinite coplanar boundary: $$\phi_0 + \phi_i = 0 \implies \phi_i = -\phi_0$$