Chapter 6, Problem 1RQ

To determine

Choose the correct option that provides whether the given equation is true or false.

Answer to Problem 1RQ

The correct option is (a) True.

Explanation of Solution

Calculation:

Consider that the Laplace and Poisson’s equations are related as follows.

$\nabla \cdot D=\nabla \cdot \epsilon E={\rho }_v$        (1)

Consider the electric field related to the electric potential by gradient relationship.

$E=-\nabla V$        (2)

Substitute Equation (2) in (1).

$\nabla \cdot D=\nabla \cdot \epsilon \left(-\nabla V\right)={\rho }_v$

$\nabla \cdot \left(-\epsilon \nabla V\right)={\rho }_v$        (3)

Equation (3) is Poisson’s equation for an inhomogeneous medium.

For homogeneous medium, Equation (3) reduces to,

${\nabla }^{2}V=-\frac{{\rho }_v}{\epsilon }$

Conclusion:

Thus, the correct option is (a) True.