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Law of Conservation of Mechanical Energy

Semicircular ...

Question

Semicircular ring of radius $0.5 m$ is uniformly charged with a total charge of $1.4 \times 10^{- 9} C$ . The electric filed intensity at centre of this ring is

A

zero

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B

320 V / m

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C

64 V / m

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D

32 V / m

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Solution

The correct option is D $32 V / m$
$\begin{matrix} A c c o r d i n g t o q u e s t i o n : r = 0.5 m q = 1.4 \times 10^{- 9} C N o w, f i n d e l e c t r i f i e l d i n t e n s i t y : θ = \frac{a r c}{r a d i u s} \Rightarrow d l = R d θ - - - - - (i) l e t, c h e o f d e n s i t y = p c h e o f t h i s e l e m e n t s = d q \Rightarrow d q = R d θ - - - - - - - - (i i) N o w, E l e t r i c f i e l d d u e t o t h e e l e m e n t s a t l a t e r : d E (m a g n i t u d e) = K d q / R^{2} d E = K d q / R^{2} cos θ i + K d q / R^{2} sin θ (- j d E = \frac{K d q}{R^{2}} [cos θ i - sin θ j] E = \frac{K p}{R} [- 2 j] N o w, w e h a v e P (c h a r g e p e r u n i t l e n g t h) = \frac{θ}{π R} s o t h a t, E = \frac{2 K}{π R^{2}} (- j) E = \frac{2 \times 9 \times 10^{9} \times 1.4 \times 10^{- 9}}{π \times {(0.5)}^{2}} ∴ E = 32 V / m s o t h e c o r r e c t o p t i o n i s D . \end{matrix}$

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