Drift Velocity

Drift velocity is a key concept in electromagnetism and solid-state physics, representing the average velocity of charged particles, like electrons, in response to an applied electric field in a conductor. While these charge carriers move randomly and collide with atoms, the electric field induces a net movement in one direction, resulting in a steady drift. Grasping drift velocity is crucial for understanding electric current flow and the behavior of conductive materials under different conditions.

1.0Definition Of Drift Velocity

• Drift Velocity $\overrightarrow{v_d}$ - It is the mean velocity acquired by the free electrons of a conductor in the inverse direction of the externally applied electric field.
• In the existence of an External field,each electron encounters a force in the opposite direction of the applied electric field.
• As the electron accelerates they frequently collide with nearby ions,between two successive collisions,an electron gains a velocity component in a direction opposite to Electric Field.
• Gain in velocity lasts for a short time and is lost in the next collision,at each collision,the electron starts afresh with a random thermal velocity.
• Relaxation Time ($\tau$): The mean time that elapses between two successive collisions of an electron. For conductors it is of order 10-14 s.

$\tau =\frac{{\tau }_1+{\tau }_2+{\tau }_3+....{\tau }_n}{N}$

2.0Derivation Of Drift Velocity

• Due to application of potential difference/Voltage across the conductor an electric field is produced inside the conductor.
• Due to the electric field, free electrons experience force opposite to the direction of the electric field.
• Due to this force these electrons start drifting in the opposite direction of the electric field and during motion they face many collisions with heavy positive ions in their path.
• So the velocity with which e^– effectively displaced is called drift velocity.
• Let us imagine at any time (t=0), any electron moving with initial velocity $V_{th}$ now due to external electric field Eit experiences force opposite to the direction of electric field E

$\overrightarrow{F_e}=e\vec{E}$ ………….1

$a_e=\frac{eE}{m}$ ………….2

Now after time $t_1$ (say) it will be having velocity $V_1$

$\overrightarrow{V_1}=\overrightarrow{V_{th}}+\frac{eE}{m}{t}_1$

Averaging over N electrons

$<\vec{V}>=<\overrightarrow{V_{th}}>+<\frac{eE}{m}{t}_1>$

$\overrightarrow{V_d}=0+\frac{e\vec{E}\tau }{m}$

$\overrightarrow{V_d}=\frac{e\vec{E}\tau }{m}$

3.0Relation Between Drift Velocity and Current

Radius r and area of cross-section A, Let n depicts the number of free electrons per unit volume or free electron density.

$V-d=\frac{dx}{dt}$ , (n= $\frac{N}{V}$=Free electron density)

dV=Adx

dN=ndV=nAdx

dq=edN=enAdx

$i=\frac{dq}{dt}=\frac{enAdx}{dt}=neA(\frac{dx}{dt})-neA{V}_d$

4.0Mobility ($\mu$)

• Ease of movement of charge carriers within a conductor known as mobility of charge carriers.
• It is defined as drift velocity per unit electric field

$\mu =\frac{V_d}{E}=\frac{1}{E}\frac{eE\tau }{m}\Rightarrow m=\frac{e\tau }{m}$

• $\mu$ is constant for a given material (If temperature is constant)
• Electrons have more mobility than protons $\left(\mu \propto \frac{1}{\text{ mass }}\right)$

5.0Formula of Current Using Drift Velocity

1. If 1 atom releases only 1 electrons

No.of atoms(N) = No. of free electrons

$\text{ No.of atoms }(N)=\text{ No. of moles }\times N_A=\frac{m}{M_W}\times N_A$

$N=\frac{m}{M_W}\times N_A$

Free electron density (n)

$n=\frac{N}{v}=\frac{m}{M_W}\frac{N_A}{m}\rho =\frac{\rho N_A}{M_W}$

Where =density of material, MW=Molecular weight

$i=nAe{V}_d$

$i=\frac{\rho N_A}{M_W}eA{V}_d$

2. $i=neA{V}_d=\frac{N}{V}eA{V}_d=\frac{N}{A\times l}eA{V}_d$

$i=\frac{Ne{V}_d}{l}\times \frac{m_e}{m_e}=\left(N\times m_e{V}_d\right)\frac{e}{m_{e}l}$

$\left(N\times m_e{V}_d\right)=\text{ Net linear momentum of all free electrons }$

$i=\frac{p_{e}e}{m_{e}l}\text{ where }{m}_e=\text{ mass of electrons, }l=\text{ length of wire }$

6.0Sample Questions On Drift Velocity

Q-1. A current I glides through a uniform wire of diameter d then the electrons drift velocity v. The same current will glides through a wire of diameter $\frac{d}{2}$ made of the same material, Find the drift velocity of the electrons in the second wire.

Solution:

$J=\frac{I}{A}=ne{V}_d$

$\frac{4I}{\pi d^2}=ne{V}_{d1}$ …………1

$\frac{16I}{\pi d^2}=ne{V}_{d2}$ ………….2

From equation 1 and 2

$\frac{4I}{16I}=\frac{V_{d1}}{V_{d2}}$

$V_{d2}=4V_{d1}$

Q-2.The no. of free electrons per 10 mm of an ordinary copper wire is 2 ✕ 1021. The average drift speed of electron is 0.25 mm/sec. Calculate the current flow in the wire.

Solution:

N=Total No. of free electrons

$i=neA{V}_d$

$i=\left(\frac{N}{Al}\right)eA{V}_d=\left(\frac{N}{l}\right)e{V}_d$

$i=\left(\frac{2\times 10^{21}}{10}\right)\times 1.6\times 10^{-19}\times 0.25=8A$

Q-3.Two wires each of radius r but of distinct materials are linked together whole (in series. If the densities of charge carrier in the two wire are in the ratio 1:4.The drift velocity of electron in the two wires will be in the ratio.

Solution:

$i=neA{V}_d$

$V_d\propto \frac{1}{n}\Rightarrow \frac{V_{d1}}{V_{d2}}=\frac{n_2}{n_1}=\frac{4n_1}{n_1}=4:1$