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Question Number 36096 by rahul 19 last updated on 28/May/18

Commented by rahul 19 last updated on 30/May/18

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Commented by tanmay.chaudhury50@gmail.com last updated on 30/May/18

$i a m t r y i n g t o s o l v e b y t h e f o l l o w i n g m e t h o d$ $1) c a l c u l a t e p o t e n t i a l f o r i n f i n i t e - v e c h a r g e$ $s h e e t a t p t h e n s u b s t r u c t i t b y t h e p o t e n t i a l$ $b y d i s c o f r a d i u s R . . . s a m e f o r + v e s h e e t a n d$ $t h e n a d d$ $2) s a m e m e t h o d f o r e l e c t r i c f i e l d . . .$ $p l s w a i t$
Commented by tanmay.chaudhury50@gmail.com last updated on 30/May/18

Commented by tanmay.chaudhury50@gmail.com last updated on 30/May/18

	Answered by tanmay.chaudhury50@gmail.com last updated on 31/May/18
	$formula E=(σ/(2ε_0 ))(1−(z/((√(z^2 +R^2 )) ))) for charge disc E=(σ/(2ε_0 )) for infinite sheet value of z=x+(l/2) for −ve charged sheet vslue of z=x−(l/2) for +ve charged sheet now E for −ve charged sheet =(((−σ))/(2ε_0 ))−(((−σ))/(2ε_0 )){1−(((x+(l/2)))/(√((x+(l/2))^2 +R^2 )))} =(((−σ))/(2ε_0 )){1−1+(((x+(l/2)))/(√((x+(l/2))^2 +R^2 )))} =(((−σ))/(2ε_0 )){(((x+(l/2)))/(√((x+(l/2))^2 +R^2 )))} similarly E for +ve charged sheet =(((+σ))/(2ε_0 )){(((x−(l/2)))/(√((x−(l/2))^2 +R^2 )))} so net E at p= (σ/(2ε_0 )){(((x−(l/2)))/(√((x−(l/2))^2 +R^2 )))−(((x+(l/2)))/(√((x+(l/2))^2 +R^2 )))} now approximatation...l≫R =(((x−(l/2)))/((x−(l/2))(√(1_ +((R/(x−(l/2))))^2 )))) ={1+((R/(x−(l/2))))^2 }^(−(1/2)) ≈1−(1/2)((R/(x−(l/2))))^2 (((x+(l/2)))/((x+(l/2))(√(1+((R/(x+(l/2))))^2 )))) =(({1+((R/(x+(l/2))))^2 }^((−1)/2) )/) ≈1−(1/2)((R/(x+(l/2))))^2 =(σ/(2ε_0 )){1−(1/2)((R/(x−(l/2))))^2 −1+(1/2)((R/(x+(l/2))))^2 } =((σR^2 )/(4ε_0 )){(((x+(l/2))^2 −(x−(l/2))^2 )/((x^2 −(l^2 /4))^2 ))} =((σR^2 )/(4ε_0 )){((4.x.(l/2))/((x^2 −(l^2 /4))^2 ))} =((σR^2 xl)/(2ε_0 (x^2 −(l^2 /4))^2 )) ≈((σR^2 xl)/(2ε_0 x^4 )) considering (x≫(l/2)) =((σR^2 l)/(2ε_0 x^3 )) pls check and suggest$
	$f o r m u l a$ $E = \frac{σ}{2 ϵ_{0}} (1 - \frac{z}{\sqrt{z^{2} + R^{2}}}) f o r c h a r g e d i s c$ $E = \frac{σ}{2 ϵ_{0}} f o r i n f i n i t e s h e e t$ $v a l u e o f z = x + \frac{l}{2} f o r - v e c h a r g e d s h e e t$ $v s l u e o f z = x - \frac{l}{2} f o r + v e c h a r g e d s h e e t$ $n o w E f o r - v e c h a r g e d s h e e t$ $= \frac{(- σ)}{2 ϵ_{0}} - \frac{(- σ)}{2 ϵ_{0}} {1 - \frac{(x + \frac{l}{2})}{\sqrt{{(x + \frac{l}{2})}^{2} + R^{2}}}}$ $= \frac{(- σ)}{2 ϵ_{0}} {1 - 1 + \frac{(x + \frac{l}{2})}{\sqrt{{(x + \frac{l}{2})}^{2} + R^{2}}}}$ $= \frac{(- σ)}{2 ϵ_{0}} {\frac{(x + \frac{l}{2})}{\sqrt{{(x + \frac{l}{2})}^{2} + R^{2}}}}$ $s i m i l a r l y E f o r + v e c h a r g e d s h e e t$ $= \frac{(+ σ)}{2 ϵ_{0}} {\frac{(x - \frac{l}{2})}{\sqrt{{(x - \frac{l}{2})}^{2} + R^{2}}}}$ $s o n e t E a t p =$ $\frac{σ}{2 ϵ_{0}} {\frac{(x - \frac{l}{2})}{\sqrt{{(x - \frac{l}{2})}^{2} + R^{2}}} - \frac{(x + \frac{l}{2})}{\sqrt{{(x + \frac{l}{2})}^{2} + R^{2}}}}$ $n o w a p p r o x i m a t a t i o n . . . l ≫ R$ $= \frac{(x - \frac{l}{2})}{(x - \frac{l}{2}) \sqrt{1_{} + {(\frac{R}{x - \frac{l}{2}})}^{2}}}$ $= {1 + {(\frac{R}{x - \frac{l}{2}})}^{2}}^{- \frac{1}{2}} \approx 1 - \frac{1}{2} {(\frac{R}{x - \frac{l}{2}})}^{2}$ $\frac{(x + \frac{l}{2})}{(x + \frac{l}{2}) \sqrt{1 + {(\frac{R}{x + \frac{l}{2}})}^{2}}}$ $= \frac{{1 + {(\frac{R}{x + \frac{l}{2}})}^{2}}^{\frac{- 1}{2}}}{}$ $\approx 1 - \frac{1}{2} {(\frac{R}{x + \frac{l}{2}})}^{2}$ $= \frac{σ}{2 ϵ_{0}} {1 - \frac{1}{2} {(\frac{R}{x - \frac{l}{2}})}^{2} - 1 + \frac{1}{2} {(\frac{R}{x + \frac{l}{2}})}^{2}}$ $= \frac{σ R^{2}}{4 ϵ_{0}} {\frac{{(x + \frac{l}{2})}^{2} - {(x - \frac{l}{2})}^{2}}{{(x^{2} - \frac{l^{2}}{4})}^{2}}}$ $= \frac{σ R^{2}}{4 ϵ_{0}} {\frac{4 . x . \frac{l}{2}}{{(x^{2} - \frac{l^{2}}{4})}^{2}}}$ $= \frac{σ R^{2} x l}{2 ϵ_{0} {(x^{2} - \frac{l^{2}}{4})}^{2}}$ $\approx \frac{σ R^{2} x l}{2 ϵ_{0} x^{4}} c o n s i d e r i n g (x ≫ \frac{l}{2})$ $= \frac{σ R^{2} l}{2 ϵ_{0} x^{3}} p l s c h e c k a n d s u g g e s t$
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