Answer
$2 \pi$
Work Step by Step
Here, we have the normal vector as: $r(t)= \lt \cos \phi \cos t, \sin t, \sin \phi \cos t \gt$ for $0 \lt t \lt 2\pi$ when $\phi$ is fixed.
$r(t)= \lt \cos \phi \cos t, \sin t, \sin \phi \cos t \gt \\\implies r'(t)= \lt -\cos \phi \sin t, \cos t, -\sin \phi \sin t \gt$
Let us suppose that $F=\lt -x, y, -z\gt$
Thus, the circulation integral can be computed as:
$\oint_C F dr=\int_0^{2 \pi} F \cdot r'(t) \ dt $
or, $= \int_0^{2\pi} \lt -\cos \phi \cos t, \sin t, -\sin \phi \cos t \gt \cdot \lt -\cos \phi \sin t, \cos t, -\sin \phi \sin t \gt \ dt$
or, $=\int_0^{2 \pi} [\cos^2 t(\sin^2 \phi+\cos^2 \phi) +\sin^2t]\ dt $
or, $=\int_0^{2 \pi} (1) \ dt $
or, $=2\pi$
