Question

In a certain two-dimensional field of force the potencial energy of a particle has the uniform `U=alphax^2+betay^2`, where `alpha` and `beta` are positive constants whose magnitudes are different. Find out:
(a) whether this field is central,
(b) what is the shape of the equipotential surfaces and also of the surfaces for which the magnitude of the vector of force `F=const.`

Answer 1

(a) We have
`F_x=-(deltaU)/(dx)=-2alphax` and `F_y=(-deltaU)/(dy)=-2betay`
So, `vecF=2alphaxveci-2betaveci` and, `F=2sqrt(alpha^2x^2+beta^2y^2)` (1)
For a central force, `vecrxxvecF=0`
Hence, `vecrxxvecF=(xveci+yvecj)xx(-2alphaxveci-2betayvecj)`
`=-2betaxyveck-2alphaxy(veck)~~0`
Hence the force is not a central force.
(b) As `U=alphax^2+betay^2`
So, `F_x=(deltaU)/(deltax)=-2alphax` and `F_y=(-deltaU)/(deltay)=-2betay`.
So, `F=sqrt(F_x^2+F_y^2)=sqrt(4alpha^2x^2+4beta^2y^2)`
According to the problem
`F=2sqrt(alpha^2x^2+beta^2y^2)=C` (constant)
or, `alpha^2x^2+beta^2y^2=(C^2)/(2)`
or, `(x^2)/(beta^2)+(y^2)/(alpha^2)=(C^2)/(2alpha^2beta^2)=k` (say) (2)
Therefore the surfaces for which F is constant is an ellipse.
For an equipotential surface `U` is constant.
So, `alphax^2+betay^2=C_0` (constant)
or, `(x^2)/(sqrt(beta^2))+(y^2)/(sqrt(alpha^2))=(C_0)/(alphabeta)=K_0` (constant)
Hence teh equipotential surface is also an ellipse.