Question

A rod is $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{m}$ long. Its charge is $-\mathbf{2}\times {\mathbf{10}}^{-\mathbf{7}}\mathbf{C}$. The observation location is $\mathbf{4}\mathbf{cm}$ from the rod, in the mid plane. In the expression

$\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{Q}}{\mathbf{r}\sqrt{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\left(\frac{L}{2}\right)}^{\mathbf{2}}}}$

what is$\mathit{r}$ in meters?

Short answer

The value of$\mathit{r}$is$\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{}\mathit{m}\mathbf{.}$

Step-by-step solution

Given data

A uniformly charged rod of length$L$.

Distance from the rod of the point where the electric field is observed is $4cm$.

Step 2:Electric field due to a uniformly charged rod

The electric field due to a rod of length$\mathit{L}$ and charge$\mathit{Q}$ uniformly distributed at a distance$\mathit{r}$ from it on its mid plane is,

role="math" localid="1668497952679" $\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{Q}}{\mathbf{r}\sqrt{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\left(\frac{L}{2}\right)}^{\mathbf{2}}}}$

Step 3:Determining the distance  r

From equation (I), it is seen that$r$is the distance of the observation point from the rod, that is,

$$\begin{gathered} r=4\text{cm} \\ =4·\left(1\text{cm}\times \frac{1\text{m}}{100\text{cm}}\right) \\ =0.04\text{m} \end{gathered}$$

Thus, $r$is meters is $0.04m$.