Question

The potential energy of a certain particle is given by $U=\frac{1}{2}(x^2-z^2)$. The force on it is

• A. $-x\hat{i}+z\hat{k}$
• B. $x\hat{i}+z\hat{k}$
• C. $\frac{1}{2}(x\hat{i}+z\hat{k})$
• D. $\frac{1}{2}(x\hat{i}-z\hat{k})$

Answer 1

The correct option is A $-x\hat{i}+z\hat{k}$

Given:
$U=\frac{1}{2}(x^2-z^2)$
Now,
$\overrightarrow{F}=-\frac{\partial U}{\partial x}\hat{i}-\frac{\partial U}{\partial z}\hat{k}$
$\frac{\partial U}{\partial x}=\frac{1}{2}\times (2x-0)=x\frac{\partial U}{\partial z}=0-(\frac{1}{2}\times 2z)=-z\Rightarrow \overrightarrow{F}=-x\hat{i}+z\hat{k}$
$$\begin{array}{c}\text{Why this question ?}\\ \text{Tips: Whenever a potential energy field is given,}\\ \text{always remember that force on the particle is given by}\\ \overrightarrow{F}=-\frac{\partial U}{\partial x}\hat{i}-\frac{\partial U}{\partial y}\hat{j}-\frac{\partial U}{\partial z}\hat{k}\end{array}$$