Case Study
Passage with linked questions
Case Set 1
Case AnalysisPassage
Riya is studying the behaviour of conduction electrons in a copper wire connected to a battery. She learns that when no electric field is applied, electrons move randomly due to thermal energy, colliding with fixed positive ions. The speed of these random motions is about 2 × 10² m/s at room temperature. When the battery is connected, an electric field is established almost instantly throughout the wire, causing each electron to acquire a small additional drift velocity superposed on its large random velocity. For a copper wire of cross-sectional area 1.0 × 10⁻⁷ m² carrying 1.5 A, with electron number density 8.5 × 10²⁸ m⁻³, the drift speed works out to about 1.1 mm/s — nearly 10⁵ times smaller than the thermal speed. Yet current appears immediately upon closing the circuit.
Question 1: Define drift velocity of electrons in a metallic conductor and state the direction in which electrons drift relative to the applied electric field.
- Drift velocity is the small average velocity acquired by free electrons in a conductor due to the applied electric field, superposed on their large random thermal velocities.
- Electrons drift in the direction opposite to the electric field (i.e., from lower to higher potential), because the field exerts a force –eE on the negatively charged electrons.
Question 2: Even though the drift speed of electrons is only about 1.1 mm/s, large currents flow in conductors. Explain why, using the expression I = nevdA.
- The current I = nevdA depends not only on drift speed vd but also on the free electron number density n, which is enormous in metals (~10²⁸–10²⁹ m⁻³).
- Even a tiny drift velocity, when multiplied by this enormous n and the cross-sectional area A, yields a substantial current; it is the vast number of charge carriers, not their individual speed, that produces large currents.
Question 3: Why is current established almost instantly when a circuit is closed, despite electrons drifting so slowly? Compare the relevant speeds quantitatively.
- When a circuit is closed, the electric field is established throughout the conductor almost instantaneously, propagating at the speed of electromagnetic waves (~3 × 10⁸ m/s), not at the electron drift speed.
- This field simultaneously acts on electrons at every point in the circuit, causing a local drift everywhere at once; current establishment does not require electrons to travel from one terminal to the other.
- The ratio of field propagation speed to drift speed is ~3 × 10⁸ / 1.1 × 10⁻³ ≈ 3 × 10¹¹, showing the field signal is about 10¹¹ times faster than electron drift, explaining the near-instantaneous response.