Concept Of Drift Velocity, Ohm’s Law and Resistance (JEE LEVEL)

J = constant, i.e.,  (I₁/A₁) = (I₂/A₂)

I₂<I₁   (as   A₂<A­₁)

If the cross-section is constant, I ∝ J. Thus, for a given cross-sectional area, the greater the current density, the greater will be the current.

Drift Velocity

The drift velocity is the average uniform velocity acquired by free electrons inside a metal conductor by the application of an electric field which is responsible for current through it.

Free electrons, under the influence of thermal agitation, move randomly  throughout the metallic conductor from one point to another in an irregular manner in all possible directions as shown in Fig. (A). However, if a source of potential difference is applied across the two ends of the conductor, the electrons gain velocities tending them to move from negative to positive side of the conductor. Average value of these velocities is termed as drift velocity. The drift velocity of the electrons is no doubt superimposed on their random velocity, but in fact this is the cause for net transportation of the charge along the conductor and hence results in a current flow.

In absence of any electric field the random motion of electrons does not contribute to any current.  The number of electrons crossing any plane from left to right is equal to the number of electrons crossing from right to left (otherwise metal will not remain equipotential).

When an electric field is applied, due to electric force the path of electrons in general becomes curved instead of straight lines and electrons drift opposite to the field

Relation between drift velocity and electric field

Due to random motion, the free electrons of metal collide with positive metal ions and undergo change in direction after every collision.  So, the thermal velocities are randomly distributed in all possible directions.Therefore, the average of all thermal velocities is zero

\vec{u}=\frac{{{{{\vec{u}}}_{1}}+{{{\vec{u}}}_{2}}+…..+{{{\vec{u}}}_{n}}}}{N}=zero

Here, {{\vec{u}}_{1}},\,{{\vec{u}}_{2}}  , . . . , {{\vec{u}}_{n}} are the individual thermal velocities of the free electrons at any given time and N is the total number of free electrons in the conductor.

However, when some potential difference V is applied across the two ends of a conductor of length l, an electric field is set up.  which is given by,           E = V / l

then  each free electron in the conductor experiences a force                \displaystyle \vec{F}=-e\vec{E} in a direction opposite to the direction of electric field.

If m is the mass of the electron, then acceleration produced is  \vec{a}=-\frac{{e\vec{E}}}{m}

At any given time, an electron has a velocity such that {{\vec{v}}_{1}}={{\vec{u}}_{1}}+\vec{a}{{\tau }_{1}} , where {{\vec{u}}_{1}}  is the thermal velocity and \vec{a}{{\tau }_{1}}  is the velocity acquired by the electron under the influence of the applied electric field.  where t₁ being the time of acceleration or time that has elapsed since the last collision.

Similarly, the velocities of the other electrons are {{\vec{v}}_{2}},\,{{\vec{v}}_{3}}…..{{\vec{v}}_{N}} , such that

{{\vec{v}}_{2}}={{\vec{u}}_{2}}+a{{\vec{\tau }}_{2}}, 

{{\vec{v}}_{3}}={{\vec{u}}_{3}}+\vec{a}{{\tau }_{3}}

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{{\vec{v}}_{n}}={{\vec{u}}_{n}}+\vec{a}{{\tau }_{N}}

The average of all these velocities of all the free electrons in the conductor is the drift velocity .Drift velocity is defined as the velocity with which the free electrons get drifted towards the positive terminal under the influence of the externally applied electric field.

So,          {{\vec{v}}_{d}}=\frac{{{{{\vec{v}}}_{1}}+{{{\vec{v}}}_{2}}+{{{\vec{v}}}_{3}}+…+{{{\vec{v}}}_{N}}}}{N}

\Rightarrow \text{      }{{\vec{v}}_{d}}=\frac{{({{{\vec{u}}}_{1}}+\vec{a}{{\tau }_{1}})+({{{\vec{u}}}_{2}}+\vec{a}{{\tau }_{2}})+….+({{{\vec{u}}}_{N}}+\vec{a}{{\tau }_{N}})}}{N}

\Rightarrow \text{      }{{\vec{v}}_{d}}=\frac{{({{{\vec{u}}}_{1}}+{{{\vec{u}}}_{2}}+….{{{\vec{u}}}_{N}})}}{N}+\vec{a}\left( {\frac{{{{\tau }_{1}}+{{\tau }_{2}}+….+{{\tau }_{N}}}}{N}} \right)

But,        \frac{{{{{\vec{u}}}_{1}}+{{{\vec{u}}}_{2}}+…+{{{\vec{u}}}_{N}}}}{N}=0

⇒  {{\vec{v}}_{d}}=\vec{a}\frac{{{{\tau }_{1}}+{{\tau }_{2}}+….+{{\tau }_{N}}}}{N}

or,           {{\vec{v}}_{d}}=\vec{a}\tau    where \tau =\frac{{{{\tau }_{1}}+{{\tau }_{2}}+….+{{\tau }_{N}}}}{N}

Here, tis called the relaxation time and it is defined as the average time of acceleration between two successive collisions.

Now if we put \vec{a}=-\frac{{e\vec{E}}}{m}  

 we get   {{\vec{v}}_{d}}=-\frac{{e\vec{E}}}{m}\tau

Current in a Conductor

Consider a conductor of uniform cross-sectional area S. If n is the number of free electrons per unit volume, the charge per unit volume is ne. If v_dis the drift velocity, electrons move a distance v_din one second. Therefore, the volume moved in one second is v_dS.

Thus, the current, \displaystyle I=\frac{{ch\arg edensity\times Volume}}{{Time}}=\frac{{(ne)\times (v{}_{d}A)}}{1}=nev{}_{d}A

J = \frac{I}{A}=ne{{v}_{d}}

or            v_d = \frac{J}{{ne}}

Metals (conductors) have large number of free electrons per unit volume (» 10²⁸/m³). Therefore, the drift velocity is very small (»10^-4 m/s) compared to random speed of electrons at room temperature (» 10⁶ m/s).

Illustration

The area of cross-section, length and density of a piece of a metal of atomic mass 60 gm/mole are 10^-6 m², 1 m and 5 x 10³ kg/m³ respectively. Find the number of free electrons per unit volume if every atom contributes one free electron. Also find the drift velocity of electrons in the metal when a current of 16 A passes through it. Given that Avogadro’s number N_A = 6 x 10²³/mole and charge on an electron e = 1.6 x 10^-19 C.

Solution

According to Avogadro’s hypothesis,

\displaystyle \frac{N}{{{{N}_{A}}}}=\frac{m}{M}

so         n = \displaystyle \frac{N}{V}={{N}_{A}}\frac{m}{{VM}}=\frac{{{{N}_{A}}d}}{M}

(as  d = \displaystyle \frac{m}{V})

n =  \displaystyle \frac{{6\times {{{10}}^{{23}}}\times \left( {5\times {{{10}}^{3}}} \right)}}{{\left( {60\times {{{10}}^{{-3}}}} \right)}} = 5 x 10²⁸ /m³

Now as each atom contributes one electron, the number of electrons per unit volume is also the same.

J = \displaystyle \frac{I}{A}=\frac{{16}}{{{{{10}}^{{-6}}}}} = 16 x 10⁶ A/m²

v_d = \displaystyle \frac{J}{{ne}}=\frac{{16\times {{{10}}^{6}}}}{{\left( {5\times {{{10}}^{{28}}}} \right)\times \left( {1.6\times {{{10}}^{{-19}}}} \right)}}=2 x 10^-3 m/s

Mobility (μ)

In a substance, although the amount of charge on a positive charge-carrier and on a negative charge–carrier may be the same, but the two types of carriers may not give equal contribution to electric current. For example, in a semiconductor, (such as Ge or Si), the two types of charge-carriers are electrons (negative) and holes (positive). When an electric field is applied, the electrons are found to drift more than the holes. We say that the mobility of electrons is more than that of holes.

Mobility (μ) of a charged particle in a substance is defined as the drift velocity per unit field applied,

m = \frac{{{{v}_{d}}}}{E}

The units of mobility are \frac{{m/\sec }}{{volt/m}} that is m²/Vs.

For a conductor we can write

J = nev_d= neμE

An intrinsic semiconductor has same number of holes and electrons, and the amount of charge on them is also same. Hence

J = Current density due to drift of electrons + Current density due to drift of holes

= J_e+ J_p = enμ_nE + epμ_pE         (μ_{n and μp}  are the mobility of negative and positive charge carriers)

J = en (μ_n + μ_p)E      (as n = p)