Answer
The answer is (b) $\rho \sin \phi = R$.
Work Step by Step
The equation of a cylinder of radius $R$ in rectangular coordinates is ${x^2} + {y^2} = {R^2}$.
The relations between rectangular and spherical coordinates are given by
$x = \rho \sin \phi \cos \theta $
$y = \rho \sin \phi \sin \theta $
$z = \rho \cos \phi $
Substituting $x$ and $y$ in ${x^2} + {y^2} = {R^2}$ gives
${\rho ^2}{\sin ^2}\phi {\cos ^2}\theta + {\rho ^2}{\sin ^2}\phi {\sin ^2}\theta = {R^2}$
${\rho ^2}{\sin ^2}\phi = {R^2}$
$\rho \sin \phi = R$
So, the answer is (b) $\rho \sin \phi = R$.
