I am doing a practice problem in my classical mechanics class and don't understand my results. The problem is to calculate the work done on a particle to move it from the origin to a distance $d$ away along the $x$ axis by a force $$\vec{F}=k \frac{x\hat i+y\hat j}{(x^2+y^2)^{3/2}}$$ where $k$ is a positive constant. I've fount that $\nabla \times \vec F=0$ so $\vec F$ is a conservative force. If I set $$V(x,y)= \frac{-k}{(x^2+y^2)^{1/2}}$$
then $\nabla V=\vec F$. So the work done is:
$$W=\int_C \vec F \cdot d\vec s=\int_C \nabla V \cdot d\vec s=V(d,0)-V(0,0)=\frac{-k}{d}+\infty$$
so the point $(0,0)$ corresponds to infinite energy? Am I doing something wrong or is there a problem with the question? I notice that the vector field $\vec F$ is undefined at $(0,0)$ so does it even make sense for a particle to travel through that point?
Best Answer
You are correct. Actually, this force field originates from a point charge located at the origin. Consequently, the vector field is only defined for points other than the origin. It doesn't make sense for a particle to travel through the origin, because that would involve going through the point charge.