Question-36096

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

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

? ?

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 \dots 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 \dots

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

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 \dots 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