An open hemispherical surface of radius \(R\) is bounded by a circular rim in the \(xy\)-plane centered at the origin, with the curved dome extending into the region \(z \ge 0\). A non-uniform magnetic field directed everywhere parallel to the \(z\)-axis is given in cylindrical coordinates by \(\vec{B} = B_0\left(\dfrac{r}{R}\right)^2 \hat{k}\), where \(r = \sqrt{x^2 + y^2}\) is the distance from the \(z\)-axis and \(B_0\) is a positive constant. Which of the following expressions is equal to the total magnetic flux \(\Phi_B\) passing outward through the curved hemispherical surface?