Hello i was wondering if anyone can help me with the following problem that I'm stuck on during my revision over Christmas period
for part a to show a force is conservative is this to just show that the curl is equal to 0 or is there an easier method?
part b and c are giving me trouble and i cant find any notes i made on how to attempt them, any help is extremely appreciated, thanks.
For part (a) your method is good enough.
Then, relative to the origin, the potential energy at point $\vec{r}$ is given by $$ U(\vec{r}) = -\int_{\vec{0}}^{\vec{r}} \mathbf{F}\cdot d\mathbf{s} $$ where any path may be chosen to go from $\vec{0}$ to $\vec{r}$; the integral is path-independent if and only if the force is conservative. So in particular, you can choose the path as having two straight-line pieces: A line from $\vec{0}$ to $\vec{r} - (\vec{r}\cdot \hat{F}) \hat{r}$ and a line from that point to $\vec{r}$. The first of those line integrals is zero becasue the line is always perpendicular to the force; the second is in the direction of the force, making the integral easy.
The result will be $$ U(\vec{r}) = - q(\vec{r}\cdot \vec{E_0}) $$
And then writing out in components, $$ U((x,y,z)) = - q(xE_x,yE_y,zE_z) \\ -\nabla U(r) = q(E_x,E_y,E_z) $$