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

To determine whether the potential ( V = 25 sin theta ) satisfies Laplace’s equation, we need to check whether it satisfies the equation ( nabla^2 V = 0 ).

In spherical coordinates, Laplace’s equation is given by:

[

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

]

Since ( V ) only depends on ( theta ) and not on ( r ) or ( phi ), we can ignore the terms involving ( r ) and ( phi ). We only need to calculate the angular part:

1. Calculate ( frac{partial V}{partial theta} ):

[

frac{partial V}{partial theta} = 25 cos theta

]

2. Calculate ( frac{partial}{partial theta} left( sin theta frac