Given the potential V = 25 sin θ, in free space, determine whether V satisfies Laplace’s equation

To determine if the given potential ( V = 25 sin theta ) satisfies Laplace’s equation in free space, we need to utilize the form of Laplace’s equation in spherical coordinates, as the potential is given in terms of (theta), which is a spherical coordinate.

Laplace’s equation in spherical coordinates (assuming azimuthal symmetry, which seems to be implied here as the potential is a function of (theta) alone) is:

[

nabla^2V = frac{1}{r^2}frac{partial}{partial r}left( r^2frac{partial V}{partial r} right) + frac{1}{r^2sintheta}frac{partial}{partial theta}left( sinthetafrac{partial V}{partial theta} right) + frac{1}{r^2sin^2theta}frac{partial^2 V}{partial phi^2} = 0

]

Given ( V = 25 sin theta ), this does not depend on ( r ) or ( phi ), so the first and third terms in the Laplace equation vanish. We’re only left with the second term:

[

frac{1}{r^2sintheta}frac{partial}{partial theta}left( sinthetafrac{