Question

A particle in a certain consevative force field has a potential energy given by $U=\frac{20xy}{z}$. The force exerted on it is

• A. $\left(\frac{20y}{z}\right)\hat{i}+\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^2}\right)\hat{k}$
• B. $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^2}\right)\hat{k}$
• C. $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}-\left(\frac{20xy}{z^2}\right)\hat{k}$
• D. $\left(\frac{20y}{z}\right)\hat{i}+\left(\frac{20x}{z}\right)\hat{j}-\left(\frac{20xy}{z^2}\right)\hat{k}$

Answer 1

The correct option is B $-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^2}\right)\hat{k}$

Given: $U=\frac{2xy}{z}$
For a conservative field, $F=-\frac{\partial U}{\partial r}$
Where,
$\therefore \overrightarrow{F}=-[\hat{i}\frac{\partial U}{\partial x}+\hat{j}\frac{\partial U}{\partial y}+\hat{k}\frac{\partial U}{\partial z}]$
$=-[\hat{i}\frac{\partial }{\partial x}\left(\frac{20xy}{z}\right)+\hat{j}\frac{\partial }{\partial y}\left(\frac{20xy}{z}\right)+\hat{k}\frac{\partial }{\partial z}\left(\frac{20xy}{z}\right)]$
$=-[\hat{i}\left(\frac{20y}{z}\right)+\left(\frac{20x}{z}\right)\hat{j}+(-\frac{20xy}{z^2})\hat{k}]$
$=-\left(\frac{20y}{z}\right)\hat{i}-\left(\frac{20x}{z}\right)\hat{j}+\left(\frac{20xy}{z^2}\right)\hat{k}$
Hence, the correct answer is option (b).