Question

In a source free region in vacuum, if the electrostatic potential, \(\phi = 2{{\rm{x}}^2} + {{\rm{y}}^2} + {\rm{c}}{{\rm{z}}^2}\), the value of constant c must be____________.

Answer 1

Concept:

In a source-free region

\({\vec \nabla ^2}\phi = 0\)

Application: As we know, In a source-free region \({\vec \nabla ^2}\phi = 0\)

\(\vec \nabla \phi = 4{\rm{x}}{{\rm{\hat a}}_{\rm{x}}} + 2{\rm{y}}{{\rm{\hat a}}_{\rm{y}}} + 2{\rm{cz}}{{\rm{\hat a}}_{\rm{z}}}\)

And \(\vec \nabla \cdot \left( {\vec \nabla \phi } \right) = 4 + {\rm{z}} + 2{\rm{c}}\)

\(\Rightarrow {\vec \nabla ^2}\phi = 6 + 2{\rm{c}} = 0\)

Thus, \({\rm{c}} = - 3\)