Question

If $\varphi =\ln \left|A\right|$ then $▽\varphi =$

• A. $\frac{A}{r^2}$
• B. $\frac{A}{r^3}$
• C. $\frac{A}{r^4}$
• D. $\frac{A}{r}$

Answer 1

The correct option is A $\frac{A}{r^2}$

$A=xi+yi+zk.$ then $\left|r\right|=\sqrt{x^2+y^2+z^2}$

and, $\varphi =\ln \left|r\right|=\frac{1}{2}\ln (x^2+y^2+z^2).$

$▽\varphi =\frac{1}{2}▽\ln (x^2+y^2+z^2)$

$=\frac{1}{2}\{i\frac{\partial }{\partial x}\ln (x^2+y^2+z^2)+j\frac{\partial }{\partial y}\ln (x^2+y^2+z^2)+k\frac{\partial }{\partial z}\ln (x^2+y^2+z^2)\}$

$=\frac{1}{2}\{i\frac{2x}{x^2+y^2+z^2}+j\frac{2y}{x^2+y^2+z^2}+k\frac{2z}{x^2+y^2+z^2}\}=\frac{xi+yj+zk}{x^2+y^2+z^2}=\frac{A}{r^2}.$