Question

What is the order of magnitude of drift velocity of electrons in a conductor ? Deduce the relation between the current flowing through a conductor and drift velocity of electrons in it.

Hint:

The two most important formulas related to drift velocity are \[ \boxed{I=neAv_d} \] and \[ \boxed{J=nev_d}. \] Also remember that the drift velocity of electrons in an ordinary conductor is very small, typically of the order of \[ \boxed{10^{-4}\ \text{m s}^{-1}}. \]

Answer 1

Concept: In a metallic conductor, a large number of free electrons are continuously moving in random directions due to thermal motion. In the absence of an electric field, these random motions cancel one another and there is no net flow of charge through the conductor. When an external electric field is applied across the conductor, each free electron experiences an electric force and acquires a small average velocity opposite to the direction of the electric field. This average velocity of the electrons is called the drift velocity. The drift velocity is extremely small compared to the random thermal velocity of electrons and is generally of the order of \[ 10^{-4}\ \text{m s}^{-1} \] to \[ 10^{-5}\ \text{m s}^{-1}. \] Thus, the order of magnitude of drift velocity is \[ \boxed{10^{-4}\ \text{m s}^{-1}}. \]

Step 1: Consider a conductor of cross-sectional area \(A\).
Let
• \(n\) be the number of free electrons per unit volume of the conductor,
• \(A\) be the cross-sectional area of the conductor,
• \(v_d\) be the drift velocity of electrons,
• \(e\) be the magnitude of charge on an electron. Suppose the electrons move with drift velocity \(v_d\). In time \(dt\), the electrons move through a distance \[ dx=v_d\,dt. \]

Step 2: Calculate the number of electrons crossing the cross-section in time \(dt\).
The volume of the conductor through which electrons pass in time \(dt\) is \[ dV=A\,dx. \] Substituting \[ dx=v_d\,dt, \] we get \[ dV=A\,v_d\,dt. \] Since \(n\) is the number of free electrons per unit volume, the total number of electrons crossing the cross-section in time \(dt\) is \[ dN=n\,dV. \] Therefore, \[ dN=nAv_d\,dt. \]

Step 3: Calculate the total charge crossing the conductor.
The charge carried by one electron is \(e\). Hence, the total charge crossing the cross-section in time \(dt\) is \[ dq=e\,dN. \] Substituting the value of \(dN\), \[ dq=e(nAv_d\,dt). \] Therefore, \[ dq=neAv_d\,dt. \]

Step 4: Obtain the expression for electric current.
By definition, \[ I=\frac{dq}{dt}. \] Substituting the value of \(dq\), \[ I=\frac{neAv_d\,dt}{dt}. \] Hence, \[ \boxed{I=neAv_d}. \] This is the required relation between the electric current and the drift velocity of electrons in a conductor. The corresponding current density is \[ J=\frac{I}{A}. \] Therefore, \[ J=\frac{neAv_d}{A}, \] or \[ \boxed{J=nev_d}. \] Thus, current in a conductor is directly proportional to the drift velocity of electrons. \[ \boxed{I=neAv_d} \] where
• \(n\) = number density of free electrons,
• \(e\) = electronic charge,
• \(A\) = cross-sectional area of the conductor,
• \(v_d\) = drift velocity.