Question

Gravitational potential in a region is given by $V=-(x+y+z)\text{J/kg}$. Find the gravitational field intensity at $(2,2,2)$.

• A. $(\hat{i}+\hat{j}+\hat{k})\text{N/kg}$
• B. $2(\hat{i}+\hat{j}+\hat{k})\text{N/kg}$
• C. $3(\hat{i}+\hat{j}+\hat{k})\text{N/kg}$
• D. $4(\hat{i}+\hat{j}+\hat{k})\text{N/kg}$

Answer 1

The correct option is A $(\hat{i}+\hat{j}+\hat{k})\text{N/kg}$

Given that,

Gravitational potential, $V=-(x+y+z)\text{J/kg}$

Let $E_g$ denote the gravitation intensity at any point,

As we know, $E_g=-[\frac{\partial V}{\partial x}\hat{i}+\frac{\partial V}{\partial x}\hat{j}+\frac{\partial V}{\partial x}\hat{k}]$

Now gravitational field intensity at $(2,2,2)$,

$\Rightarrow E_{(2,2,2)}=-\left[-\hat{i}-\hat{j}-\hat{k}\right]=(\hat{i}+\hat{j}+\hat{k})\text{N/kg}$

Hence, option (a) is correct.

$$\begin{array}{c}\text{Why this question?}\\ \text{To familiarize students with the}\\ \text{relationship between gravitational}\\ \text{intensity gravitational potential.}\end{array}$$