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.`

In a certain two-dimensional field of force the potencial energy of a

1.	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» (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.

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