MP Board · Class 12 · Physics · Current ElectricityDefine drift velocity of free electrons in a conductor. Establish the relation between electric current and drift velocity, and hence deduce Ohm's law at the microscopic level.

Definition of Drift Velocity

• Drift Velocity ($v_d$): It is defined as the average velocity with which free electrons get drifted towards the positive terminal of a conductor under the influence of an applied external electric field.
• The magnitude of drift velocity is typically of the order of $10^{-4}\text{ m/s}$.
• Formula: $v_d = \frac{eE}{m} \tau$, where $e$ is electronic charge, $E$ is electric field, $m$ is mass of electron, and $\tau$ is relaxation time.

Relation Between Electric Current and Drift Velocity\nConsider a uniform conductor of length $L$ and cross-sectional area $A$. Let:

• $n$ = number density of free electrons (number of free electrons per unit volume)
• $e$ = charge of an electron
• $v_d$ = drift velocity of electrons \nTotal volume of the conductor of length $v_d$ and area $A$ is $A v_d$. The number of electrons crossing any cross-section of the conductor in time $t=1\text{ second}$ is given by: $$\text{Number of electrons} = n A v_d$$ \nTotal charge $q$ flowing across the cross-section in time $t$ is: $$q = (n A v_d t) e$$ \nSince electric current $I$ is the rate of flow of charge ($I = \frac{q}{t}$), we get: $$I = \frac{n A v_d t e}{t}$| $$I = n e A v_d$$

Deduction of Ohm's Law\nWe know that drift velocity is given by:

$$v_d = \frac{e E \tau}{m}$| \nSubstituting the expression for $v_d$ into the current equation: $$I = n e A \left( \frac{e E \tau}{m} \right)$| $$I = \frac{n e^2 A \tau E}{m}$| \nSince the electric field $E$ in terms of potential difference $V$ across length $L$ is $E = \frac{V}{L}$, substitute this: $$I = \frac{n e^2 A \tau}{m} \left( \frac{V}{L} \right)$| \nRearranging the terms to isolate $V$: $$V = \left( \frac{m L}{n e^2 A \tau} \right) I$$ \nThe term inside the parenthesis $\left( \frac{m L}{n e^2 A \tau} \right)$ depends only on the material properties, dimensions, and temperature of the conductor, which remains constant. This term is defined as the electrical resistance ($R$) of the conductor: $$R = \frac{m L}{n e^2 A \tau}$| \nThus, we obtain: $$V = I R$$ or $\frac{V}{I} = R$ \nThis is the microscopic deduction of Ohm's law.