English
Hindi
1. Electric Charges and Fields
normal

બે બિંદુવત ઘન વીજભારને $d$ અંતરે રાખેલા છે.એક ત્રીજા ઘન વીજભારને લંબદ્વિભાજક પર $x$ અંતરે મૂકેલા છે,$x$ ના કયા મૂલ્ય માટે લાગતું બળ મહતમ થાય?

A

$x = \frac{d}{{\sqrt 2 }}$

B

$x = \frac{d}{2}$

C

$x = \frac{d}{{2\sqrt 2 }}$

D

$x = \frac{d}{{2\sqrt 3 }}$

Solution

$F_{net} = 2F\ cos \theta$

$F = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{Qq}}{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}}$

$\cos \theta  = \frac{x}{{\sqrt {{x^2} + \frac{{{d^2}}}{4}} }}$

${F_{net}} = 2 \times \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{Qq}}{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}} \times \frac{x}{{{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}^{1/2}}}}$$ = \frac{{2Qqx}}{{4\pi {\varepsilon _0}{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}^{3/2}}}}$

$\frac{{d{F_{net}}}}{{dx}} = 0$

$\frac{d}{{dx}}\left[ {\frac{{2Qqx}}{{4\pi {\varepsilon _0}{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}^{3/2}}}}} \right] = 0$

$\left[ {{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}^{ – 3/2}} – 3{x^2}{{\left( {{x^2} + \frac{{{d^2}}}{4}} \right)}^{ – 5/2}}} \right] = 0$

$x =  \pm \frac{d}{{2\sqrt 2 }}$

Standard 12
Physics

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