Question

Graph the magnitude of the full expression for the field $\mathit{E}$ of a rod along the midplane vs. $\mathit{r}$. Does $\mathit{E}$fall off monotonically(with distance)?

Short answer

The plot of $\mathit{E}$vs $\mathit{r}$ is

$\mathit{E}$falls off monotonically with distance.

Step-by-step solution

Given data

A uniformly charged rod of length with an electric field is being measured at a distance$r$ from it along the midplane.

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="1668497036664" $\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{\mathbf{Q}}{\mathbf{r}\sqrt{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}{\left(\frac{L}{2}\right)}^{\mathbf{2}}}}$.....(i)

Step 3:Determination of the plot of the electric field

Plot the electric field defined in equation (i) as a function of $r$as follows:

As can be seen from the plot, the electric field falls of monotonically with distance.