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To find the Laplace equation value of the given potential field ( V = rho cos phi + z ), we need to check if it satisfies the Laplace equation, which is given by:
[
nabla^2 V = 0
]
In spherical coordinates, the Laplacian operator is given by:
[
nabla^2 V = frac{1}{rho^2} frac{partial}{partial rho} left( rho^2 frac{partial V}{partial rho} right) + frac{1}{rho^2 sin phi} frac{partial}{partial phi} left( sin phi frac{partial V}{partial phi} right) + frac{1}{rho^2} frac{partial^2 V}{partial z^2}
]
Given ( V = rho cos phi + z ):
1. Calculate ( frac{partial V}{partial rho} ):
[
frac{partial V}{partial rho} = cos phi
]
2. Calculate ( frac{partial^2 V}{partial rho^2} ):
[
frac{partial^2 V}{partial rho^2} = 0
]
3.
Answer: a
Explanation: (Del)2(ρ cosφ + z)= (cos φ/r) – (cos φ/r) + 0
= 0, this satisfies Laplace equation. The value is 0.