Question

In a certain two-dimensional field of force, the potential energy of a particle have the form $U=\alpha x^2+\beta y^2$, where $\alpha$ and $\beta$ are positive constants whose magnitudes are different. Find out what is the shape of the equipotential surfaces and also of the surfaces for which the magnitude of the vector of force $\overrightarrow{F}=$ constant.

Answer 1

Potential energy of particle has form $U=\alpha x^2+\beta y^2$

For equipotential surfaces, $U=\alpha x^2+\beta y^2=constant$

Thus it has a shape of an ellipse.

Force acting on the particle=$F=-\frac{dU}{dr}$

where $r=x\hat{i}+y\hat{j}$

Thus $F=2\alpha x+2\beta y=constant$

Hence the planes $2\alpha x+2\beta y=constant$ have constant force acting on them.