Question

. The potential field 𝑉 = 2𝑥

2𝑦𝑧 − 𝑦

3

𝑧 exists in a dielectric medium having 𝜀 = 2𝜀0 (a) Does V satisfy

Laplace`s equation? (b) calculate the total charge within the unit cube 0 < 𝑥, 𝑦, 𝑧 < 1𝑚.

Answer 1

Answer

a) for laplace condition

$\nabla ^2V=0$ ......... eq1

$\nabla ^2V=\frac{d^2V}{dx^2}+\frac{d^2V}{dy^2}+\frac{d^2V}{dz^2}$

$V=2x^2yz-y^3z$

$\frac{d^2V}{dx^2}=4yz$

$\frac{d^2V}{dy^2}=-6yz$

$\frac{d^2V}{dz^2}=0$

Putting all value in equation 1

$\nabla ^2V\ne0$

So this potential does not satisfy.

b) using poission equation

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

Charge

$Q=\int\rho dv$

Integrate by putting charge density

$Q=\epsilon_0=8.85\times10^{-12}C$