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 $-x\hat{i}+z\hat{k}$

$F_x=\frac{-\partial U}{\partial x}=\frac{-\partial }{\partial x}\left(\frac{x^2-z^2}{2}\right)=-x$
$F_z=\frac{-\partial U}{\partial z}=\frac{-\partial }{\partial z}\left(\frac{x^2-z^2}{2}\right)=z$
$\therefore \overrightarrow{F}=-x\hat{i}+z\hat{k}$