91Ó°ÊÓ

1. Particle 1 of charge$+4e$ is above a floor by distance,${\mathrm{d}}_1=2\text{″¾³¾´Ç°ù}0.002\text{″¾}$
2. Particle 2 of charge$+6e$ is on the floor, at distance${\mathrm{d}}_2=6\text{″¾³¾´Ç°ù}0.006\text{″¾}$ horizontally from particle 1.

Using the given charges in the formula of force from Coulomb's law, we can get the x-component of force on particle 2 due to particle 1. To get the x-component of the force, we can get the unit-vector of the distance.

Formulae:

The magnitude of the electrostatic force between any two particles,

${\mathrm{F}}_{21}=\frac{\left|{\mathrm{q}}_1{\mathrm{q}}_2\right|}{4{\mathrm{Ï€Î\mu }}_0{\mathrm{r}}^2},\mathrm{where}\mathrm{r}=\sqrt{{\mathrm{d}}_1^2+{\mathrm{d}}_2^2}$ (i)

The distance vector in unit-vector notation, $\widehat{\mathrm{r}}=\frac{\mathrm{r}}{\stackrel{→}{\mathrm{r}}}=\frac{{\mathrm{d}}_2\widehat{\mathrm{i}}−{\mathrm{d}}_1\widehat{\mathrm{j}}}{\sqrt{{\mathrm{d}}_1^2+{\mathrm{d}}_2^2}}$ (ii)

Since both$q_1$and$q_2$are positively charged, particle 2 is repelled by particle 1, so the direction of ${\mathrm{F}}_{21}$is away from particle 1 and toward 2.

In unit-vector notation, the force acting on the particle 2 due to 1 can be given as:$\stackrel{→}{{\mathrm{F}}_{12}}={\mathrm{F}}_{12}\widehat{\mathrm{r}}$

Now, for the x-component of the force, we can take the $\hat{i}$vector value of the distance from equation (ii) as given:

$\begin{array}{c}{\stackrel{→}{\mathrm{F}}}_{12\left(\mathrm{x}\right)}=\frac{{\mathrm{F}}_{12}{\mathrm{d}}_2}{\sqrt{{\mathrm{d}}_1^2+{\mathrm{d}}_2^2}}\\ =\mathrm{k}\frac{{\mathrm{q}}_1{\mathrm{q}}_2{\mathrm{d}}_2}{{({\mathrm{d}}_1^2+{\mathrm{d}}_2^2)}^{3/2}}\text{(substitutingequation(i))}\\ =\frac{(9×{10}^9\frac{\text{N}â‹\dots {\text{m}}^{\text{2}}}{{\text{C}}^{\text{2}}})(4×1.60×{10}^{−19}\text{ C})(6×1.60×{10}^{−19}\text{ C})(6.00×{10}^{−3}\text{″¾})}{{[{(2.00×{10}^{−3}\text{″¾})}^2+{(6.00×{10}^{−3}\text{″¾})}^2]}^{\frac{3}{2}}}\\ =1.31×{10}^{−22}\text{ N}\end{array}$

Hence, the x-component of the force is$1.31×{10}^{−22}\text{ N}$