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(D) The magnetic field produced by an infinitely long straight wire carrying current $I$ at a distance $r$ from the wire is given by $B = \frac{\mu_0

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(D) The magnetic field produced by an infinitely long straight wire carrying current $I$ at a distance $r$ from the wire is given by $B = \frac{\mu_0 I}{2 \pi r}$.In this problem,the wire lies along the $Y$-axis. The magnetic field lines are concentric circles in the $xz$-plane,centered on the $Y$-axis.The circular loop is in the $xy$-plane. The magnetic field vector $\vec{B}$ at any point on the loop lies in the $xz$-plane (specifically,it is perpendicular to the $Y$-axis and the radial vector from the wire).The area vector $\vec{A}$ of the loop in the $xy$-plane is directed along the $Z$-axis (i.e.,$\vec{A} = A \hat{k}$).Since the magnetic field $\vec{B}$ is always in the $xz$-plane and the area vector $\vec{A}$ is along the $Z$-axis,the magnetic field $\vec{B}$ is always perpendicular to the area vector $\vec{A}$ at every point on the loop.Therefore,the magnetic flux $\phi_B = \int \vec{B} \cdot d\vec{A} = \int B dA \cos(90^\circ) = 0$.

An infinitely long wire lying along the $Y$-axis carries a current $I$ as shown in the figure. What is the magnetic flux through a circular loop of radius $R$ in the $xy$-plane? [Assume $\mu_0$ is the magnetic permeability of free space.]

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