# Electric Current

The **current density** ($J$) in a material is defined as the current ($I$) flowing through it per unit area ($A$). It is proportional to the electric field $E$ inside the material. The constant of proportionality is called **electrical conductivity** ($\sigma$) of the material. Thus,
\begin{align}
J=\frac{I}{A}=\sigma E.
\end{align}

The drift velocity of the electrons is the average speed of the electrons inside the conductor. It is given by
\begin{align}
v_d=\frac{I}{neA}
\end{align}
where $n$ is the number of electrons per unit volume.

## Problems from IIT JEE

**Problem (IIT JEE 1997):**
A steady current flows in a metallic conductor of non-uniform cross-section. The quantity/quantities constant along the length of the conductor is (are),

- current, electric field and drift speed
- drift speed only
- current and drift speed
- current only

**Solution: **
In steady state, charge can not accumulate in a conductor. Thus, constant current $i={\mathrm{d}q}/{\mathrm{d}t}$ flows along the length of conductor. The current density at a point having cross-section area $A$ is given by $J={i}/{A}$. The current density is related to electric field by $J=\sigma E$, where conductivity $\sigma$ is a material property. The drift velocity $v_d$ is given by $i=neAv_d$. Thus, both $E$ and $v_d$ are inversely proportional to $A$.

## Related

- Ohm's Law
- Electrical Resistance
- Kirchhoff's Laws
- Heating Effect of Current

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