Question

11. II A small glass bead charged to $+6.0\mathrm{nC}$is in the plane that bisects a thin, uniformly charged, $10-\mathrm{cm}$-long glass rod and is $4.0\mathrm{cm}$from the rod's center. The bead is repelled from the rod with a force of $840\mathrm{\mu N}$. What is the total charge on the rod?

Short answer

Total charge on the rod is$39.84\mathrm{nC}$.

Step-by-step solution

Formula for force

A charged bead $+6\mathrm{nC}$is$4\mathrm{cm}$distant from a rod with a length of $10\mathrm{cm}$.

A repulsive force of bead islocalid="1649133963049" width="94">840mu/Nonexerted by Rod.

This indicates that the charge on the road is positive, as opposite charges repel each other.

Force due to a rod on any charge is given by

$\mathrm{F}=\frac{\mathrm{kQq}}{\mathrm{r}\sqrt{{\mathrm{r}}^2+\frac{{\mathrm{l}}^2}{4}}}$

localid="1649134063653" $\mathrm{Q}$is charge on rod.

$\mathrm{q}$is charge of bead.

$\mathrm{r}$is distance between bead and rod.

$\mathrm{l}$is length of rod.

Calculation for charge on the rod

Charge is,

$\mathrm{Q}=\frac{\mathrm{Fr}\sqrt{{\mathrm{r}}^2+\frac{{\mathrm{l}}^2}{4}}}{\mathrm{kq}}$

$=\frac{\left(840\times {10}^{-6}\mathrm{N}\right)(0.04\mathrm{m})\sqrt{(0.04\mathrm{m}{)}^2+{\left(\frac{0.1\mathrm{m}}{2}\right)}^2}}{\left(9\times {10}^9\mathrm{N}·{\mathrm{m}}^2/{\mathrm{C}}^2\right)\left(6\times {10}^{-9}\mathrm{C}\right)}$

$=39.84\mathrm{nC}$