CUET UG Physics Booster Test – 2 Electron Dynamics and Drift

In a metallic conductor, free electrons undergo frequent collisions with the fixed positive ions of the crystal lattice. These collisions are essentially elastic in nature. As a result, the magnitude of the electron's thermal speed remains approximately the same before and after collision, but its direction becomes completely random. Since collisions occur in all possible directions, there is no preferred direction of motion in thermal equilibrium. This randomization is responsible for the zero average velocity of electrons when no external electric field is applied. The NCERT drift velocity model assumes that collisions continuously randomize the direction of electron motion while maintaining the thermal speed distribution. Therefore, after a collision, the electron has nearly the same speed but a random direction.

In a conductor at thermal equilibrium, free electrons are in continuous random motion due to thermal energy. Whenever they collide with the fixed positive ions of the lattice, their directions are randomized. Consequently, there is no preferred direction of electron motion at any instant. Since equal numbers of electrons move in opposite directions, the vector average of all electron velocities becomes zero: 1/N∑v_i=0 Therefore, statements 2, 3, and 4 are correct. Statement 1 is incorrect because random thermal motion does not produce a steady electric current. Although electrons move rapidly due to thermal energy, the motion is completely random and the contributions cancel each other. A net current appears only when an external electric field establishes a drift velocity superimposed on the random thermal motion.

Although individual electrons possess very high thermal speeds, their directions are completely random. For every electron moving in one direction, there is statistically another moving in the opposite direction. Therefore, the average velocity is v_(avg)=0 This does not mean electrons are stationary; rather, their motions cancel out when averaged over a large number of particles. Hence statements A, C, and D correctly describe thermal equilibrium. Statement B is incorrect because high speeds alone do not imply a non-zero average velocity.

Electric current is associated with the net directed motion of charge carriers. In the absence of an external electric field, electrons possess only random thermal motion. Since the average velocity is zero, v_(avg)=0 the drift velocity is also zero. Current is given by I=neAv_d Since v_d=0, I=0 Therefore, no macroscopic electric current flows through the conductor even though electrons continue moving randomly at high thermal speeds.

According to Newton's second law, F=ma Therefore, a=F/m Substituting the given values: a=1.456×10^(-16)/9.1×10^(-31)a=1.60×10^(14)  m/s^2 Electrons possess extremely small mass, so even a modest electric force can produce enormous acceleration. In real conductors, however, frequent collisions prevent electrons from continuously accelerating and instead lead to a steady drift velocity.

When oppositely charged discs are attached to the ends of a metallic cylinder, an electric field is established inside the conductor. Free electrons present in the metal experience an electric force due to this field and begin to accelerate opposite to the field direction. As electrons move, they gradually neutralize the charge imbalance present at the ends. This motion constitutes a temporary electric current. However, because no external source continuously replenishes the charges, the internal electric field decreases with time and eventually becomes zero. Once the field disappears, the drift motion stops and only random thermal motion remains. Thus, the current exists only for a short duration and is not steady.

After a collision, an electron possesses a random thermal velocity v_i. Under the influence of an electric field, it accelerates with acceleration a. Using V_i=v_i+at_i where V_i is the instantaneous velocity before the next collision. Rearranging, t_i=V_i-v_i/a This expression gives the elapsed time since the electron's last collision. It forms the basis for introducing the relaxation time concept used in deriving the drift velocity formula.

Relaxation time τis defined as the average time interval between successive collisions of electrons with lattice ions. Mathematically, τ=⟨t_i⟩ Hence statement 1 is correct. Since τis an average, some electrons will have elapsed times larger than τwhile others will have smaller values, making statement 3 correct. Statement 4 is also correct because relaxation time characterizes the average interval between randomizing collisions. Statement 2 is incorrect because drift velocity remains finite: v_d=eEτ/m and does not become infinite.

The instantaneous velocity of an electron can be expressed as V_i=v_i-eE/mt_i where: v_i is the random thermal velocity immediately after collision. V_i is the instantaneous velocity before the next collision. -eE/mt_i is the velocity gained due to the electric field. Averaging over all electrons gives the drift velocity v_d. Thus: 1-d,  2-a,  3-c,  4-b which corresponds to Option A.

The instantaneous velocity is V_i=v_i-eE/mt_i Taking the average over all electrons, v_d=⟨v_i⟩-eE/m⟨t_i⟩ Since random thermal velocities average to zero, ⟨v_i⟩=0 and ⟨t_i⟩=τ Therefore, v_d=-eE/mτ The negative sign indicates that electron drift occurs opposite to the electric field direction.

When an electric field is applied across a metallic conductor, free electrons accelerate under the influence of the electric force. However, they do not continue accelerating indefinitely because they frequently collide with the fixed positive ions of the crystal lattice. After each collision, much of the directed motion gained from the field is lost and the electron's motion becomes randomized again. The electric field then accelerates the electron once more until the next collision occurs. This continuous cycle of acceleration and randomization produces a constant average drift velocity rather than continuously increasing velocity. Thus, the observed steady drift speed is a consequence of repeated collisions with lattice ions. This explanation forms the basis of the classical drift velocity model discussed in NCERT.

When a circuit is closed, the electric field propagates through the conductor at a speed comparable to that of electromagnetic signals. Since free electrons are already present throughout the entire conductor, there is no need for electrons to travel from the battery to distant parts of the circuit before current begins. As soon as the field is established, electrons at every location experience an electric force and begin drifting. Therefore, current appears throughout the circuit almost simultaneously. This explains why electrical devices respond nearly instantaneously even though individual electrons drift very slowly.

The quantity n represents the number of free charge carriers per unit volume of a conductor. For metals, its value is typically of the order of 10²⁸–10²⁹ m⁻³. Because this number is extremely large, even a very small drift velocity produces a significant current. During a time interval Δt, electrons contained in a cylindrical region of length |v_d|Δt and cross-sectional area A cross the chosen area. The number of such electrons equals nA∣v_d∣Δt This result forms the basis of the microscopic current equation I=neA∣v_d∣ Hence statements 1, 3 and 4 are correct, while statement 2 is incorrect because n is defined per unit volume, not per unit area.

In a conductor carrying current, electrons possess an average drift velocity v_d. During a small time interval Δt, only those electrons lying within a cylindrical region of cross-sectional area A and length ∣v_d∣Δt can cross the chosen area. The number of electrons crossing the area is nA∣v_d∣Δt where n is the number density of free electrons. Since each electron carries charge -e, the total charge transported is ΔQ=-neA∣v_d∣Δt The current is defined as I=ΔQ/Δt or in magnitude, I=neA∣v_d∣ Therefore, the microscopic current model requires electrons to move only a distance ∣v_d∣Δt during the interval Δt. Statement D is incorrect because it wrongly suggests that electrons must travel distances much larger than ∣v_d∣Δt. The derivation of current does not require any such larger displacement.

The microscopic expression for electric current is I=neA∣v_d∣ where n is the number density of free electrons, e is the electronic charge, A is the cross-sectional area, and v_d is the drift velocity. For a conductor placed in an electric field E, the magnitude of drift velocity is ∣v_d∣=eEτ/m where τis the relaxation time and m is the mass of an electron. Substituting this value into the current equation, I=neA(eEτ/m)I=ne^2EτA/m Current density is defined as current per unit cross-sectional area: j=I/A Therefore, j=ne^2Eτ/m This is the microscopic form of Ohm's law. Comparing it with j=σE gives the conductivity σ=ne^2τ/m Hence Option B is correct.

Conductivity is given by σ=ne^2τ/m Substituting, σ=(8.5×10^(28))(1.6×10^(-19))^2(2.5×10^(-14))/9.1×10^(-31) Since (1.6×10^(-19))^2=2.56×10^(-38)σ=(8.5)(2.56)(2.5)×10^(28-38-14)/9.1×10^(-31)σ=54.4×10^(-24)/9.1×10^(-31)σ≈5.98×10^7 S m^(-1) This value is close to the conductivity of good conductors such as copper.

Mobility is defined as μ=∣v_d∣/E It measures how easily charge carriers move through a material under the influence of an electric field. Using the drift velocity relation, v_d=eEτ/m we obtain μ=eτ/m Thus mobility depends directly on relaxation time and inversely on mass. By convention, mobility is taken as a positive quantity representing the magnitude of drift velocity per unit electric field.

Mobility is defined as μ=∣v_d∣/E The SI unit of drift velocity is m s^(-1) and the SI unit of electric field is V m^(-1) Therefore, μ=m s^(-1)/V m^(-1)=m^2/V s Since mobility is defined using the magnitude of drift velocity, it is treated as a positive quantity.

To estimate the conduction electron density of copper, the number of copper atoms per unit volume is first calculated. Copper has a density of approximately 9.0×10^3 kg m^(-3) and one mole of copper (6.0×10^(23) atoms) has a mass of about 63.5 g. Using these values, the number of atoms present in one cubic meter can be calculated. Since each copper atom contributes approximately one conduction electron, the electron number density is obtained from the atomic density. Thus statements 1, 2 and 4 are correct. Statement 3 is incorrect because copper is generally taken to contribute one free electron per atom, not two.

Free electrons in a conductor possess random thermal motion even in the absence of an external electric field. Their typical thermal speed is of the order of 2×10^2 m s^(-1) which corresponds to 1-c. When an electric field is applied, electrons acquire a small average drift velocity. This drift speed is extremely small, typically around 10^(-3) m s^(-1) which corresponds to 2-b. Although the drift speed is very small, electrical effects are transmitted rapidly because the electric field propagates through the conductor at a speed close to 3×10^8 m s^(-1) which corresponds to 3-a. In thermal equilibrium without an applied field, electron motion is completely random and the vector average velocity becomes 0 m s^(-1) which corresponds to 4-d. Therefore, the correct matching is: 1-c,  2-b,  3-a,  4-d