The average speed of free electrons in a conductor
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| Random Motion of Electrons |
A conductor contains a large number of free electrons. All electrons are in motion due to thermal energy and collision takes place with the heavy fixed ions. After the collision, the speed of the electron is the same but in random directions. In this way, the average velocity of free electrons is zero.
Acceleration of Electron in Electric Field Formula
Drift Velocity Derivation
When an electric field is applied to the conductor, free electrons accelerate.
We know that
`F = ma` ................eq. (1)
Force in an electric field
`F = - e E` ................eq. (2)
From eq (1) and (2)
`ma = - e E`
`a = - \frac {e E}{m}`
According to the first equation of motion.
`v = u + a t`
If `v = v_d`, `u = 0` `a = - \frac { e E }{m}` and `t = \tau` then
`v_d = 0 + ( - \frac { e E }{m}) \tau`
`v_d = - \frac { e E }{m} \tau`
`v_d = \frac { e E }{m} \tau`
Where,
e = Charge on the electron.
E = Intensity of electric field.
`\tau =` Relaxation time (the average time between successive collisions).
`v_d =` Drift velocity.
u = 0 ( velocity immediately after the last collision).
v = velocity at time t.
Drift Velocity `v_d`
In a conductor free electrons move with an average velocity that is independent of time, although electrons are accelerated. This average velocity is called the drift velocity.
Conductivity of a Conductor Formula Derivation
Consider a cylindrical conductor of length l and cross-sectional area A in electric field E.
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| Flow of Current in Metallic Conductor |
Consider a planar area A, located inside the conductor such that the normal to the area is parallel to electric field E. Free electrons in the conductor experience a force in the opposite direction of the applied electric field.
If n is the number of free electrons per unit volume in the metal.
Then
Total number of electrons in a conductor
`= n V`
`= n A l`
`= n A v_d \Delta t`
Total charge transported across this area A in the opposite to the electric field
`Q = - e n A v_d \Delta t`
Total charge transported across this area A in the direction of the electric field
`Q = + e n A v_d \Delta t`
`I \Delta t = + e n A v_d \Delta t`
`I = + e n A v_d `
`I = + e n A \frac {e E}{m} \tau ` `\because v_d = \frac { e E }{m} \tau`
`I = e n A \frac {e E}{m} \tau `
`I = \frac {n A e^2 \tau }{m} E`
`\frac {I}{A} = \frac {n e^2 \tau }{m} E`
`J = \frac {n e^2 \tau }{m} E` `(J = \frac {I}{A} = \text{current density})`
In vector form
`\vec J = \frac {n e^2 \tau }{m} \vec E`
`\vec J = \sigma \vec E`
Here,
` \frac {n e^2 \tau }{m} = \sigma` = conductivity
It is clear that electrical conduction produces Ohm's law but remember that there is an assumption that `\tau` and n are constants and independent of E.
Conclusion
* Free electrons in a conductor move randomly due to thermal energy, resulting in zero average velocity.
* When an electric field is applied, electrons accelerate, creating a net drift velocity (`v_d`).
* The drift velocity formula is `v_d = \frac{e E \tau}{m}`.
* This result in the current density `J = \frac{n e^2 \tau}{m} E`
* Conductivity (`\sigma`) is given by `\sigma = \frac{n e^2 \tau}{m} `
* Electrical conduction follows Ohm's law when `\tau` and `n` are constant.
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