Question

If G = $20r\widehat{a_{\varphi }},$ then $\oint \overrightarrow{G}.\overrightarrow{dl}$ over the circular path r = 2 m, $\theta ={30}^{\circ },0<\varphi <2\pi$ will be

• A. $60\pi$
• B. 80
• C. $80\pi$
• D. 60

Answer 1

The correct option is C $80\pi$

For spherical coordinate systems,

$\overrightarrow{dl}=r\sin \theta d\varphi \widehat{a_{\varphi }}$

$\oint \overrightarrow{G}.\overrightarrow{dl}={\int }_0^{2\pi }\left(20r\widehat{a_{\varphi }}\right)\left(r\sin \theta d\varphi \widehat{a_{\varphi }}\right)$

$=20.r^2.sin\theta \left(2\pi \right)$

$=20\left(2{)}^{2}sin{30}^{\circ }\right(2\pi )$

$=80\pi$