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[NOTES/EM-07001]-Electric Current

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In this section we compute the leading term in the magnetic field of a current loop at large distances and obtain an expression for the magnetic moment of the loop.


 

1. Current Density 

The electric current is a flow of electric charges, the electrons to be precise. Most common place where this appears is the current in wires that are laid for connecting different points for specific usages such as lighting or fans. The current in wire is amount of charge flowing per second across a point on the wire. For later uses we need work with charges whose flow is not restricted to wires in a circuit. Consider charge distributed over a volume and moving in different directions. Such a situation is described by the charge density $\rho(\vec{r},t)$ and velocity of charges $\vec{v}(\vec{r},l)$. If we have a small surface area $\Delta S$, Fig 1.(a), with normal along unit vector $\hat{n}$, we may ask for current flowing through the surface $\Delta S$. This current is defined as the amount of charge crossing the surface per sec


The charges have velocity $\vec{v}$ and will more a distance $\vec{v}\Delta t$ in time $\Delta t$. The charges that cross a small surface $\Delta S$ in time $\Delta t$ will be those contained in a volume of cylinder of base $\Delta S$ and having length $l=v\Delta t$, the axis of cylinder being parallel to the velocity $\vec{v}$, \Figref{cu}(b) If the normal to the base, $\hat{n}$, volume of the cylinder will be $\Delta S\cos\theta(\vec{v}\Delta t)$ where $\theta$ is the angle between $\hat{n}$ the normal $\hat{n}$ and the velocity $\vec{v}$. Thus the amount of charge crossing $\Delta S$ per second is $\Delta S(v\Delta t)\rho\cos\theta$. We can write this as charge crossing $\Delta S$ per sec \(= \Delta S(\rho v)\cos\theta\Delta t=\Delta S(\rho\vec{v})\hat{n}\cdot\Delta t\) \(=\vec{j}\cdot\overrightarrow{\Delta S}\cdot \Delta t\) \noindent where $\vec{j}=\rho\vec{v}, \overrightarrow{\Delta S}=\hat{n}S$.\\ The quantity $\vec{j}=\rho\vec{v}$ is called current density and it gives charge flowing through a small surface $\Delta S$ per sec as $(\hat{n}\cdot\vec{j})\Delta S$. For an arbitrary surface one will need to do a surface integral of $(\hat{n}\cdot\vec{j})\Delta S$.

2. Current in Thin Wire

Note that every point in wire the velocity of charge carriers \(\vec{v}\),hence the current density \(\vec{j}=\rho \vec{v}\), line element \(d\ell\) and the normal to the cross section are all parallel, being parallel to the tangent vector. Therefore we need not carry vector signs for these objects. We first need to get relation between the current density \(\vec{j}\) and the current flowing \(I d\ell\) in the wire having the cross section \(A\). Consider a small line element \(d\ell\) of the wire. Corresponding volume element will be Considering charges in the small line element will be volume $dV=A\,d\ell$, see \Figref{cu}(c), the charge flowing through the cross section of the wire per sec will \(j dV= jA(d\ell)\). The number of charge carriers in this element will be \(NdV = N A (d\ell)\). $\overrightarrow{d\ell}$ and the current density is related to the electric current $I$ in the wire by $$ \vec{J}\,dV=I\overrightarrow{d\ell}\,\,. $$ This result is easily verified using the definitions of $\vec{J}$ and $I$ \vspace{0.2in} \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 $$ \rho=Nq\qquad\qquad \vec{J}=Nq\vec{v} $$
References

1. Sec. 5.1.3 Current David Griffiths, Introduction to Electrodynamics, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi, (2002).

2. Sec. 13-2 Electric current; Conservation of charge R. P. Feynman, Robert B. Leighton and Mathew Sands Lectures on Physics, vol-II, B.I. Publications (1964) 

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