use, for a current-carrying conductor, the expression I = Anvq , where n is the number density of charge carriers

Electric Current

Electric current is the rate at which charge flows through a conductor. It is defined as the amount of charge \$Q\$ passing a given cross‑section per unit time \$t\$:

\$I = \frac{dQ}{dt}\$

Microscopic Expression for Current

For a uniform conductor carrying a steady current, the current can be expressed in terms of the properties of the charge carriers:

\$I = A\,n\,v\,q\$

• \$I\$ – electric current (A)
• \$A\$ – cross‑sectional area of the conductor (m²)
• \$n\$ – number density of charge carriers (m⁻³)
• \$v\$ – drift speed of the carriers (m s⁻¹)
• \$q\$ – charge of each carrier (C)

Derivation of \$I = Anvq\$

Consider a segment of conductor of length \$L\$ and cross‑section \$A\$. In a time interval \$t\$, charge carriers travel a distance \$v t\$. The volume of material swept out is \$A v t\$, which contains \$n A v t\$ carriers. Each carrier carries charge \$q\$, so the total charge that passes through the cross‑section in time \$t\$ is

\$\Delta Q = n A v t \, q\$

Dividing by \$t\$ gives the current:

\$I = \frac{\Delta Q}{t} = A n v q\$

Physical Meaning of Each Parameter

Application: Determining Drift Speed

Re‑arranging the expression gives the drift speed:

\$v = \frac{I}{A n q}\$

Example

1. Find the drift speed of electrons in a copper wire of radius \$1.0\ \text{mm}\$ carrying a current of \$5.0\ \text{A}\$.
2. Data: copper has \$n \approx 8.5\times10^{28}\ \text{m}^{-3}\$, \$q = -e = -1.60\times10^{-19}\ \text{C}\$.

Solution:

\$A = \pi r^{2} = \pi (1.0\times10^{-3}\ \text{m})^{2} = 3.14\times10^{-6}\ \text{m}^{2}\$

\$v = \frac{5.0\ \text{A}}{(3.14\times10^{-6}\ \text{m}^{2})(8.5\times10^{28}\ \text{m}^{-3})(1.60\times10^{-19}\ \text{C})}\$

\$v \approx 1.2\times10^{-4}\ \text{m s}^{-1}\$

The drift speed is only a few hundred micrometres per second, illustrating why the electric signal propagates much faster than the individual carriers.

Factors Affecting Current

• Cross‑sectional area (\$A\$): Larger area allows more carriers to flow, increasing \$I\$ for a given \$n\$, \$v\$, \$q\$.
• Number density (\$n\$): Materials with more free carriers (metals) conduct better than insulators.
• Drift speed (\$v\$): Proportional to the electric field; \$v = \mu E\$, where \$\mu\$ is mobility.
• Charge per carrier (\$q\$): Determines the magnitude of charge transported per carrier.

Suggested Diagram

Key Points to Remember

• The macroscopic current \$I\$ is the product of geometric, material, and dynamical factors: \$I = A n v q\$.
• Drift speed is typically very small; the rapid propagation of electric signals is due to the electric field, not carrier motion.
• Understanding each term helps in analysing how changes in material or dimensions affect circuit behaviour.