$\vec{v}$ is a conservative vector field such that $$ \vec{v} = f(r)(x,y), \text{ where } r=\sqrt{x^2+y^2} $$ $f:\mathbb{R}^+\to \mathbb{R}$ and is continuously differentiable
- Work out the scalar potential of $\vec{v}$ involving $f$
- Find the anti clockwise flux $$\oint_{C_{R}} \vec{v} \cdot n ds$$ in terms of $f$ and find all $f$ such that the flux is independent of $R$. $C_R$ is the circle centered at (0,0) with radius $R$
I know how to work this out without $f$ complicating things, I just don't know how to apply the same methods with $f$, e.g when finding the scalar potentials and setting the scalar potential equal to the integral of each component of $\vec{v}$ how would you integrate the expression involving $f$.
Sorry if I have not explained things too well, any help is appreciated!
Note that we have
$$\begin{align} \oint_{C_R}\vec v \cdot \hat n\,ds&=\int_0^{2\pi}\left.\left(f(r)\left(\hat xx+\hat yy\right)\frac{\hat xx+\hat yy}{r}\,r\right)\right|_{r=R}\,d\theta\\\\ &=2\pi R^2f(R) \end{align}$$
is independent of $R$ if and only if $\displaystyle f(r)=\frac{A}{r^2}$ for some constant $A$.
To find the scalar potential, $\Phi$, for $\vec v$, we solve the equation $\nabla \Phi =\vec v$. In polar coordinates, $(r,\theta)$, this becomes
$$\frac{\partial \Phi}{\partial r}=rf(r)\implies \Phi(r)=\int^r r'f(r')\,dr'$$