91Ó°ÊÓ

(a).

Figure is,

(a).

$+2\mathrm{nC}$at A causes a force:

${\mathrm{F}}_1=\frac{1}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}×\frac{\left(2\mathrm{nc}\right)\mathrm{P}}{{\left(\mathrm{x}×{10}^{-2}\right)}^2}$

Force resulting from$-4\mathrm{nC}$at A:

${\mathrm{F}}_2=\frac{1}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}×\frac{\left(4\mathrm{nc}\right)\mathrm{P}}{{\left.(\mathrm{x}-1)×{10}^{-2}\right)}^2}$

At${\mathrm{PF}}_{\text{net}}=0$,

${\mathrm{F}}_1={\mathrm{F}}_2$

localid="1649573924219" role="math" $\frac{1}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}×\frac{\left(2\mathrm{nc}\right)\mathrm{P}}{{\left(x×{10}^{-2}\right)}^2}=\frac{1}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}×\frac{(-4\mathrm{nc})\mathrm{P}}{{\left((x-1)×{10}^{-2}\right)}^2}$

localid="1649573963479" role="math" $$\begin{gathered} \frac{1}{x^2}=\frac{2}{(x-1{)}^2} \\ (x-1{)}^2=2x^2 \\ {x}^2-2x+1=2x^2 \\ {x}^2+2x-1=0 \end{gathered}$$

solve the above quadratic equation by quadratic equation, we get

$x=\frac{-bÂ\pm \sqrt{b^2-4ac}}{2a}$where, role="math" localid="1649574059895" $a=1,b=2,c=-1$

role="math" localid="1649574190259" $$\begin{gathered} ⇒x=\frac{-2Â\pm \sqrt{4-(4×1×(-1\left)\right)}}{2\left(1\right)} \\ ⇒x=\frac{-2Â\pm \sqrt{4+4}}{2} \\ ⇒x=\frac{-2Â\pm \sqrt{8}}{2} \\ ⇒x=\frac{-2Â\pm 2\sqrt{2}}{2} \\ ⇒x=-1Â\pm \sqrt{2} \end{gathered}$$

$∴$The two real roots are

$x=-1-1.414=-2.414cm;x=-1+1.414=0.414cm$

From the above solution, the x-coordinate in which the proton placed is$-2.4cm$

(b). we need to verify the net force to be zero. $∴$Forces acting on electrons must be equal.

Force is,

${\mathrm{F}}_1={\mathrm{F}}_2$

localid="1649575589767" role="math" $$\begin{gathered} ⇒\frac{1}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}×\frac{\left(2\mathrm{n}\right)\mathrm{e}}{{(-2.4)}^2}=\frac{1}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}×\frac{\left(4\mathrm{n}\right)\mathrm{e}}{(-3.4{)}^2} \\ ⇒\frac{0.346ne}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}}=\frac{0.346ne}{4{\mathrm{Ï€Ï\mu }}_{\mathrm{o}}} \end{gathered}$$

$∴$The two forces acting on electrons are equal.

Yes. The net force between electrons within the same place is zero.