Question

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

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

Answer 1

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

$V=-(x+y+z)J/kg$

Relation between gravitational intensity $\overrightarrow{G}$ and gravitational potential:

$\overrightarrow{G}=-\overrightarrow{\nabla }V=-(\frac{\partial }{\partial x}\hat{i}+\frac{\partial }{\partial y}\hat{j}+\frac{\partial }{\partial z}\hat{k})(-[x+y+z])=\hat{i}\frac{\partial }{\partial x}(x+y+z)+\hat{j}\frac{\partial }{\partial y}(x+y+z)+\hat{k}\frac{\partial }{\partial z}(x+y+z)=\hat{i}+\hat{j}+\hat{k}$

Therefore the field is constant everywhere