[NOTES/EM-07015] Current Density --- Examples

Example 1: Current in a thin wire

At every point in a wire, the velocity of charge carriers \(\vec{v}\), hence the current density \(\vec{j}=\rho \vec{v}\), the line element \(d\ell\) and the normal to the cross section are all parallel to the tangent vector. Therefore, we need not carry vector signs for these objects. Consider a small length \(d\ell\) of the wire as shown in \Figref{cu}(c). If the cross section of the wire is \(A\), we have \begin{equation} j dV = j A (d\ell) =I d\ell. \end{equation} The last step follows from definition of the current density which implies that \(jA\) is the charge flowing per sec through the area \(A\) and is, therefore, the current \(I\). \noindent Usually the charges that made up the current are point particles and then $\rho=Nq$, where $N$ is the number of charge carriers per unit volume and $q$ is the charge on the particles. Thus $$ \rho=Nq, \qquad\qquad \vec{J}=Nq\vec{v} $$ If the number of charge carriers per unit volume is \(N\), the current density will be given by \(j=\rho v = qN v\). The current \(I\) in a wire is the charge flowing across a cross section per sec and is equal to \(Nq v A\) The number of charge carriers in this line element will be \(NdV = N A (d\ell)\). \FName

Example 2: Rotating Disk

A disk having uniform charge density $\sigma$ is rotating about its own axis with angular velocity. The surface current density will be given by \begin{align*} \vec{K} = & \sigma\vec{v} = \sigma\vec{r}\times\vec{\omega}\\ \end{align*}

Example 3: Rotating Sphere

A sphere carries a uniform charge $\rho$ per unit volume and rotates about $z$ axis passing through the origin. the amount of charge contained in small volume $\Delta V$ at position $\vec{r}$ is $\rho\Delta V$ and its velocity is $\vec{\omega}\times\vec{r}$. Therefore the current density is

\begin{eqnarray}
\vec{J}\Delta V &=& (\rho\vec{\omega}\times\vec{r})\Delta V \\
\therefore \qquad \bar{J} &=& \rho(\bar{\omega}\times\bar{r})\Delta V
\end{eqnarray}