What causes the erratic motion of electrons within a conductor?

The random motion of electrons in a conductor is caused by collisions between free electrons and the ions within the material.

Why do free electrons in a metallic conductor always drift in the direction opposite to that of the applied electric field?

This occurs because the force exerted on the negatively charged free electrons by the electric field is always in the opposite direction to that of the electric field itself.

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Drift Velocity

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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 $v_{d}$ - 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 ( $τ$ ): The mean time that elapses between two successive collisions of an electron. For conductors it is of order 10-14 s.

$τ = \frac{τ _{1} + τ _{2} + τ _{3} + .... τ _{n}}{N}$

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 $V_{t h}$ now due to external electric field Eit experiences force opposite to the direction of electric field E

$F_{e} = e E$ ………….1

$a_{e} = \frac{e E}{m}$ ………….2

Now after time $t_{1}$ (say) it will be having velocity $V_{1}$

$V_{1} = V_{t h} + \frac{e E}{m} t_{1}$

Averaging over N electrons

$< V >=< V_{t h} > + < \frac{e E}{m} t_{1} >$

$V_{d} = 0 + \frac{e E τ}{m}$

$V_{d} = \frac{e E τ}{m}$

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.

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

dV=Adx

dN=ndV=nAdx

dq=edN=enAdx

$i = \frac{d q}{d t} = \frac{e n A d x}{d t} = n e A (\frac{d x}{d t}) - n e A V_{d}$

4.0Mobility ( $μ$ )

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

$μ = \frac{V _{d}}{E} = \frac{1}{E} \frac{e E τ}{m} \Rightarrow m = \frac{e τ}{m}$

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

5.0Formula of Current Using Drift Velocity

If 1 atom releases only 1 electrons

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

$No.of atoms (N) = No. of moles \times N_{A} = \frac{m}{M _{W}} \times N_{A}$

$N = \frac{m}{M _{W}} \times N_{A}$

Free electron density (n)

$n = \frac{N}{v} = \frac{m}{M _{W}} \frac{N _{A}}{m} ρ = \frac{ρ N _{A}}{M _{W}}$

Where =density of material, MW=Molecular weight

$i = n A e V_{d}$

$i = \frac{ρ N _{A}}{M _{W}} e A V_{d}$

$i = n e A V_{d} = \frac{N}{V} e A V_{d} = \frac{N}{A \times l} e A V_{d}$

$i = \frac{N e V _{d}}{l} \times \frac{m _{e}}{m _{e}} = (N \times m_{e} V_{d}) \frac{e}{m _{e} l}$

$(N \times m_{e} V_{d}) = Net linear momentum of all free electrons$

$i = \frac{p _{e} e}{m _{e} l} where m_{e} = mass of electrons, l = length of wire$

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:

$J = \frac{I}{A} = n e V_{d}$

$\frac{4 I}{π d ^{2}} = n e V_{d 1}$ …………1

$\frac{16 I}{π d ^{2}} = n e V_{d 2}$ ………….2

From equation 1 and 2

$\frac{4 I}{16 I} = \frac{V _{d 1}}{V _{d 2}}$

$V_{d 2} = 4 V_{d 1}$

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

$i = n e A V_{d}$

$i = (\frac{N}{A l}) e A V_{d} = (\frac{N}{l}) e V_{d}$

$i = (\frac{2 \times 1 0 ^{21}}{10}) \times 1.6 \times 1 0^{- 19} \times 0.25 = 8 A$

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:

$i = n e A V_{d}$

$V_{d} \propto \frac{1}{n} \Rightarrow \frac{V _{d 1}}{V _{d 2}} = \frac{n _{2}}{n _{1}} = \frac{4 n _{1}}{n _{1}} = 4 : 1$

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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 $v_{d}$ - 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 ( $τ$ ): The mean time that elapses between two successive collisions of an electron. For conductors it is of order 10-14 s.

$τ = \frac{τ _{1} + τ _{2} + τ _{3} + .... τ _{n}}{N}$

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 $V_{t h}$ now due to external electric field Eit experiences force opposite to the direction of electric field E

$F_{e} = e E$ ………….1

$a_{e} = \frac{e E}{m}$ ………….2

Now after time $t_{1}$ (say) it will be having velocity $V_{1}$

$V_{1} = V_{t h} + \frac{e E}{m} t_{1}$

Averaging over N electrons

$< V >=< V_{t h} > + < \frac{e E}{m} t_{1} >$

$V_{d} = 0 + \frac{e E τ}{m}$

$V_{d} = \frac{e E τ}{m}$

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.

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

dV=Adx

dN=ndV=nAdx

dq=edN=enAdx

$i = \frac{d q}{d t} = \frac{e n A d x}{d t} = n e A (\frac{d x}{d t}) - n e A V_{d}$

4.0Mobility ( $μ$ )

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

$μ = \frac{V _{d}}{E} = \frac{1}{E} \frac{e E τ}{m} \Rightarrow m = \frac{e τ}{m}$

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

5.0Formula of Current Using Drift Velocity

If 1 atom releases only 1 electrons

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

$No.of atoms (N) = No. of moles \times N_{A} = \frac{m}{M _{W}} \times N_{A}$

$N = \frac{m}{M _{W}} \times N_{A}$

Free electron density (n)

$n = \frac{N}{v} = \frac{m}{M _{W}} \frac{N _{A}}{m} ρ = \frac{ρ N _{A}}{M _{W}}$

Where =density of material, MW=Molecular weight

$i = n A e V_{d}$

$i = \frac{ρ N _{A}}{M _{W}} e A V_{d}$

$i = n e A V_{d} = \frac{N}{V} e A V_{d} = \frac{N}{A \times l} e A V_{d}$

$i = \frac{N e V _{d}}{l} \times \frac{m _{e}}{m _{e}} = (N \times m_{e} V_{d}) \frac{e}{m _{e} l}$

$(N \times m_{e} V_{d}) = Net linear momentum of all free electrons$

$i = \frac{p _{e} e}{m _{e} l} where m_{e} = mass of electrons, l = length of wire$

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:

$J = \frac{I}{A} = n e V_{d}$

$\frac{4 I}{π d ^{2}} = n e V_{d 1}$ …………1

$\frac{16 I}{π d ^{2}} = n e V_{d 2}$ ………….2

From equation 1 and 2

$\frac{4 I}{16 I} = \frac{V _{d 1}}{V _{d 2}}$

$V_{d 2} = 4 V_{d 1}$

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

$i = n e A V_{d}$

$i = (\frac{N}{A l}) e A V_{d} = (\frac{N}{l}) e V_{d}$

$i = (\frac{2 \times 1 0 ^{21}}{10}) \times 1.6 \times 1 0^{- 19} \times 0.25 = 8 A$

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:

$i = n e A V_{d}$

$V_{d} \propto \frac{1}{n} \Rightarrow \frac{V _{d 1}}{V _{d 2}} = \frac{n _{2}}{n _{1}} = \frac{4 n _{1}}{n _{1}} = 4 : 1$

Frequently Asked Questions

What causes the erratic motion of electrons within a conductor?

Why do free electrons in a metallic conductor always drift in the direction opposite to that of the applied electric field?

Frequently Asked Questions

What causes the erratic motion of electrons within a conductor?

Why do free electrons in a metallic conductor always drift in the direction opposite to that of the applied electric field?

Join ALLEN!

Drift Velocity

1.0Definition Of Drift Velocity

2.0Derivation Of Drift Velocity

3.0Relation Between Drift Velocity and Current

4.0Mobility ( $μ$ )

5.0Formula of Current Using Drift Velocity

6.0Sample Questions On Drift Velocity

Table of Contents

Join ALLEN!

Drift Velocity

1.0Definition Of Drift Velocity

2.0Derivation Of Drift Velocity

3.0Relation Between Drift Velocity and Current

4.0Mobility ( $μ$ )

5.0Formula of Current Using Drift Velocity

6.0Sample Questions On Drift Velocity

Table of Contents

Frequently Asked Questions

What causes the erratic motion of electrons within a conductor?

Why do free electrons in a metallic conductor always drift in the direction opposite to that of the applied electric field?

Frequently Asked Questions

What causes the erratic motion of electrons within a conductor?

Why do free electrons in a metallic conductor always drift in the direction opposite to that of the applied electric field?

Join ALLEN!

Drift Velocity

1.0Definition Of Drift Velocity

2.0Derivation Of Drift Velocity

3.0Relation Between Drift Velocity and Current

4.0Mobility (μ)

5.0Formula of Current Using Drift Velocity

6.0Sample Questions On Drift Velocity

Table of Contents

Join ALLEN!

Drift Velocity

1.0Definition Of Drift Velocity

2.0Derivation Of Drift Velocity

3.0Relation Between Drift Velocity and Current

4.0Mobility (μ)

5.0Formula of Current Using Drift Velocity

6.0Sample Questions On Drift Velocity

Table of Contents

4.0Mobility ( $μ$ )

4.0Mobility ( $μ$ )