Question

Gravitational potential due to a mass distribution is given by $V=(x^{2}y+y^{2}z)$. The gravitaional field can be expressed as

• A. $-[2xy\hat{i}+(x^2+2yz)\hat{j}+y^2\hat{k}]$
• B. $-(x^2\hat{i}+y^2\hat{j}+z^2\hat{k})$
• C. $-(x\hat{i}+y^{}\hat{j}+z^{}\hat{k})$
• D. $-[2xy\hat{i}+yz\hat{k}+zx\hat{k}]$

Answer 1

The correct option is A $-[2xy\hat{i}+(x^2+2yz)\hat{j}+y^2\hat{k}]$

$E_x=\frac{-\partial V}{\partial x}=-2xy{E}_y=\frac{-\partial V}{\partial y}=-(x^2+2yz)$ [As $E=\frac{-\partial V}{\partial r}$]

$E_z=\frac{-\partial V}{\partial z}=-y^{2}E=(-2xy)\hat{i}-(x^2+2yz)\hat{j}-\left(y^2\right)\hat{k}$