Question

find a scalar potential energy function \(U(x,y,z)\) so that \[\nabla U = \left\langle \frac{-kx}{\left( x^2+y^2+z^2 \right)^{3/2}}\, ,\, \frac{-ky}{\left( x^2+y^2+z^2 \right)^{3/2}}\, ,\, \frac{-kx}{\left( x^2+y^2+z^2 \right)^{3/2}} \right\rangle\]

Answer 1

so I'm not sure how I could go about this without a huge amount of algebra. First of all I was wondering if I am allowed to factor the denominator out to show that the vector field is conservative like this...\[\nabla U = \frac{-k}{\left( x^2+y^2+z^2 \right)^{3/2}}\left\langle x\,,\,y\,,\,z\right\rangle\]and then\[\frac{\partial U_1}{\partial y}=0=\frac{\partial U_2}{\partial x}\]\[\frac{\partial U_1}{\partial z}=0=\frac{\partial U_3}{\partial x}\]\[\frac{\partial U_2}{\partial z}=0=\frac{\partial U_3}{\partial y}\] so \(\text{curl}\,\nabla U=0\)

Answer 2

After that is there any tricks I can use to save pages of writing when trying to figure this out?

Answer 3

\[\large \nabla U=<U_x, U_y, U_z>\]\[\large U_x=\frac{ -kx }{ (x^2+y^2+z^2)^{3/2} } \rightarrow U=\int\limits_{}^{} \frac{ -kx }{ (x^2+y^2+z^2)^{3/2} }dx +g(y,z)\]
problem: \[\large \left| U \right|=\sqrt{U _x^2+U_y^2+U_z^2}\]

Answer 4

is the third component of \[\large \nabla U\]\[\large \frac{k\color{red}x}{(x^2+y^2+z^2)^{3/2}}\] really kx and not kz?

Answer 5

oops yes it is kz. typo

Answer 6

so I integrate that wrt x and then I know that my constant of integration is a function of y and z only right?

Answer 7

so now I take the partial of that function wrt y?

Answer 8

sorry i'm just doing that now but I have \[U=\frac{k}{\left( x^2+y^2+z^2\right)^{1/2}}+h(y,z)\] and I am going to find \(U_y\)

Answer 9

yep

Answer 10

which I get \[U_y=\frac{-ky}{\left( x^2+y^2+z^2\right)^{3/2}}+\frac{\partial g(y,x)}{\partial y}\]

Answer 11

or \[\large \frac{ \partial g }{ \partial y }=0\] which means g(y,z) is a function of z only

Answer 12

right ok got that.

Answer 13

computing Uz, we conclude that \[\large \frac{dg}{dz}=0\] so g(z) is a numeric constant

Answer 14

alright I just arrived there as well!

Answer 15

therefore \[\large U=\frac{ k }{ (x^2+y^2+z^2)^{1/2} }+C\] the scalar potential function.

i was wrong earlier in \[\large \left| U \right|=\sqrt{U_x^2+U_y^2+U_z^2}\]

Answer 16

ok thanks a lot. I really appreaciate it. I understand how to do this clearly now!

Answer 17

YW