When a solid sphere moves through a liquid, the liquid opposes the motion with a force $F$. The magnitude of $F$ depends on the coefficient of viscosity $\eta$ of the liquid, the radius $r$ of the sphere and the speed $v$ of the sphere. Assuming that $F$ is proportional to different powers of these quantities, guess a formula for $F$ using the method of dimensions.

Let $\left[{\in }_0\right]$ denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then :

Stoke's law states that the viscous drag force $F$ experienced by a sphere of radius $a$a, moving with a speed $v$ through a fluid with coefficient of viscosity $\eta$, is given by $F=6\pi \eta av$.

If this fluid is flowing through a cylindrical pipe of radius $r$, length $l$ and a pressure difference of $P$ across its two ends, then the volume of water $V$ which flows through the pipe in time $t$ can be written as, $\frac{V}{t}=k{\left(\frac{P}{l}\right)}^a{\eta }^b{r}^c$, where $k$ is a dimensionless constant. Correct values of $a, b$ and $c$ are-