Question

The magnitude Jof the current density in a certain lab wire with a circular cross section of radius R = 2.00 mmis given by $\mathit{J}\mathbf{=}\left(3.00\times {10}^8\right){\mathit{r}}^{\mathbf{2}}$, with Jin amperes per square meter and radial distance rin meters. What is the current through the outer section bounded by r = 0.900 Rand r = R ?

Short answer

Current through the outer section bounded by radial distance r = 0.900 R and r = R is$2.59\times {10}^{-3}A$ .

Step-by-step solution

The given data

a) Current density,$J\left(r\right)=\left(3\times {10}^8\right)r^2$

b) Radius of the wire,$R=2mm$

c) Bounded radial distances,$r=0.9R$ and$r=R$

Understanding the concept of the flow of current

The current is the rate of flow of charges per unit of time. The current density is the current per unit cross-section area of the rate of flow of charges per unit time per unit area.

We can use the relation between the current and the current density to find the current through the outer section bounded between the two radii.

Formulae:

The equation of the current flowing through a small area,$i=\int \overrightarrow{J}.\overrightarrow{dA}$ ...(i)

Here, i is current,$\overrightarrow{J}$ is current density,$\overrightarrow{dA}$ is the area of cross-section.

The cross-sectional area of the circle, $A={\mathrm{\pi r}}^2$ ...(ii)

Calculation of the current through the bounded outer section

We have, the given value of the current density as:$J\left(r\right)=\left(3\times {10}^8\right)r^2$

The differential cross-sectional area value using equation (ii) can be given as follows:

$dA=2\mathrm{\pi r}\mathrm{dr}$

Substituting these above values in the equation (i), we can get the contained current within the width of the concentric ring assumingis directed along the wire for the radial distance varying from$r=0.900R$to r = R as follows:

$$\begin{gathered} i=\int \left(3\times {10}^8\right)r^{2}2\mathrm{\pi rdr} \\ =2\mathrm{\pi }\left(3\times {10}^8\right){\int }_{\mathrm{r}=0.900}^{\mathrm{r}=\mathrm{R}}{\mathrm{r}}^3\mathrm{dr} \\ =2\mathrm{\pi }\left(3\times {10}^8\right){\left[\frac{{\mathrm{r}}^4}{4}\right]}_{0.900\mathrm{R}}^{\mathrm{R}} \\ =\frac{\mathrm{\pi }\left(3\times {10}^8\right)}{2}{\left[{\mathrm{r}}^4\right]}_{0.900\mathrm{R}}^{\mathrm{R}} \\ =\frac{1}{2}\left(3\times {10}^8\right)\left[{\mathrm{R}}^4-{\left(0.900\mathrm{R}\right)}^4\right] \\ =\frac{1}{2}\left(3\times {10}^8\right)\left[{\mathrm{R}}^4-0.6561{\mathrm{R}}^4\right] \\ =\frac{1}{2}\left(3\times {10}^8\right)\left[0.3439\right]{\mathrm{R}}^4 \\ =\frac{1}{2}\left(3\times {10}^8\right)\left[0.3439\right]{\left(0.002\right)}^4\mathrm{A}\left(\because \mathrm{R}=0.002\mathrm{m}\right) \\ =2.59\times {10}^{-3}\mathrm{A} \end{gathered}$$

Hence, the value of the current is$2.59\times {10}^{-3}\mathrm{A}$ .