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Question Number 46292 by Umar last updated on 23/Oct/18

A + v e p o i n t c h a r g e o f m a g n i t u d e q i s l o c a t e d o n t h e y - a x i s

a t a p o i n t o f y = + d a n d a n o t h e r - v e c h a r g e o f s a m e m a g n i t u d e

i s l o c a t e d o n t h e p o i n t y = - d .

A t h i r d + v e c h a r g e o f s a m e m a g n i t u d e i s l o c a t e d

a t s o m e p o i n t o n x - a x i s .

(1) - w h a t i s t h e m a g n i t u d e a n d d i r e c t i o n o f t h e f o r c e o n

t h i r d c h a r g e ?

(a) - w h e n i t s l o c a t e d o n t h e o r i g i n .

(b) - w h e n i t s c o o r d i n a t e i s x .

(2) - s h o w t h a t w h e n x i s l a r g e c o m p a r e d t o d i s t a n c e d,

t h e f o r c e i n (b) i s i n v e r s e l y p r o p o r t i o n a l t o t h e

c u b e o f t h e d i s t a n c e f r o m o r i g i n .

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Oct/18

a) F o r c e o n 3 r d c h a r g e b y c h a r g e A i s r e p u l s i v e

f o r c e o n 3 r d c h a r g e b y c h a r g e B i s a t t r c t i v e

n e t f o r c e o n 3 r d c h a r g e w h e n i t i s a t o r i g i n

i s F = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{d^{2}} + \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{d^{2}} = 2 \times \frac{1}{4 π ϵ_{0}} \times \frac{q^{2}}{d^{2}}

d i r e c t i o n i s - v e y a x i s

v e c o r i c a l l y

{\vec{F}}_{1} = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{d^{2}} (- \vec{j})

{\vec{F}}_{2} = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{d^{2}} (- \vec{j})

n e t f o r c e = {\vec{F}}_{1} + {\vec{F}}_{2} = 2 \times \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{d^{2}} (- \vec{j})

b) {\vec{F}}_{1} = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{{(d^{2} + x^{2})}^{\frac{3}{2}}} (- d j + x i)

{\vec{F}}_{2} = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{{(d^{2} + x^{2})}^{\frac{3}{2}}} (- d j - x i)

{\vec{F}}_{R} = {\vec{F}}_{1} + {\vec{F}}_{2} = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{{(d^{2} + x^{2})}^{\frac{3}{2}}} (- 2 d j)

n e t f o r c e = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{{(d^{2} + x^{2})}^{\frac{3}{2}}} \times 2 d d i r e c e c t i o n - v e y a x i s

F_{R} = \frac{1}{4 π ϵ_{0}} \frac{q^{2}}{{x^{2} (1 + \frac{d^{2}}{x^{2}})}^{\frac{3}{2}}} \times 2 d \approx \frac{1}{4 π ϵ_{0}} \times \frac{q^{2} \times 2 d}{x^{3}} [\frac{d}{x} \to 0 a s x >> d]

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Commented by Umar last updated on 23/Oct/18

t h a n k y o u s i r

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