Electron Drift Velocity: Definition, Formula, Derivation and Numerical Examples & 50-Objective Questions with Explanations

⚡ Electron drift velocity is an important concept in electrical engineering. It explains how free electrons move through a conductor when an electric field is applied. In this article, we will understand drift velocity from the basics, derive its formula step by step, study its relationship with electric current and current density, and solve numerical problems.

📑 Contents

1. Beginner’s Vocabulary: Important Words Explained
2. Definition of Electron Drift Velocity
3. Movement of Electrons Inside a Conductor
4. Direction of Electron Drift and Conventional Current
5. Drift Velocity Formula: Step-by-Step Derivation
6. Meaning of Symbols and SI Units
7. Relationship Between Drift Velocity and Current Density
8. Solved Numerical Example 1
9. Solved Numerical Example 2
10. Drift Velocity in Terms of Electric Field
11. Electron Mobility
12. Current Density, Conductivity and Resistivity
13. Factors Affecting Electron Drift Velocity
14. Drift Velocity vs Random Velocity
15. Drift Velocity vs Electrical Signal Speed
16. Important Formula Summary
17. Remember Points for Competitive Exams
18. 50 Important Objective Questions on Electron Drift Velocity — Detailed Solutions

📖Important Words Explained

Before continuing, understand these words. Learning the meaning of each term makes the formulas much easier to follow.

• Charge: A basic property of matter that causes electrical forces. Charge is measured in coulombs (C).
• Electron: A negatively charged particle found in atoms. In metals, some electrons can move through the material.
• Conductor: A material, such as copper, that allows electric charge to move relatively easily.
• Voltage (potential difference): The electrical energy transferred per unit charge between two points. Voltage is measured in volts (V). A battery provides a potential difference that can establish an electric field in a connected circuit.
• Electric field: Force per unit positive charge at a point. Its SI unit is newtons per coulomb (N/C), equivalent to volts per metre (V/m).
• Drift: A gradual average movement in one preferred direction. Here, it means the average directed motion of charge carriers caused by an electric field.
• Velocity: The rate of change of position, including direction. Speed tells how fast something moves; velocity also specifies its direction. Drift velocity is an average velocity, not the random velocity of a single electron.
• Current: The rate at which electric charge passes a chosen cross-section. Its SI unit is the ampere (A), where 1 A = 1 C/s.
• Current density: Current flowing through each unit of area perpendicular to the current. Its SI unit is A/m².
• Conductance: A measure of how easily a particular component allows current to flow for a given voltage. Conductance is the reciprocal of resistance: G = 1/R. Its SI unit is the siemens (S).
• Conductivity: A material property describing how readily the material conducts current. It is represented by σ and measured in S/m. Conductance belongs to a component and depends on its geometry as well as its material; conductivity is a material property.
• Resistance: The opposition a component offers to current. It is measured in ohms (Ω).
• Resistivity: A material property related to how strongly the material opposes current. It is represented by ρ and measured in Ω·m. For a uniform conductor, σ = 1/ρ.
• Cross-sectional area: The area of a cut across the conductor, perpendicular to its length. It is represented by A and measured in m².
• Number density (n): The number of mobile charge carriers in each cubic metre. For electron conduction, it is commonly expressed in m⁻³.
• SI units: The International System of Units, the standard measurement system used in science and engineering.
• Mean relaxation time (τ): In the classical Drude model, a characteristic average time scale over which electron momentum is relaxed by scattering.
• Mobility (μ): Drift-velocity magnitude per unit electric-field magnitude in the linear-response model. Its SI unit is m²/(V·s).

Important distinction: Conductance and conductivity are related but are not interchangeable. For a uniform conductor of length L and cross-sectional area A, G = σA/L. Thus, conductance depends on the conductor's dimensions, while conductivity describes the material.

📚 1. What Is Electron Drift Velocity?

A conductor is a material that allows electric charge to move through it easily. Copper and aluminium are common examples of conductors used in electrical wiring.

Inside a metal conductor, many electrons are available to move. These are called free electrons because they are not permanently attached to a particular atom and can move through the material.

Even when no voltage is applied, free electrons are in continuous, random motion due to their thermal energy. However, their random motion does not produce a sustained net flow of electric charge through the conductor.

When a voltage is applied across the conductor, an electric field is established inside it. The electric field exerts a force on the free electrons. The electrons continue to collide with atoms and other scattering centres, but their motion develops a small average velocity in a preferred direction.

This average velocity is called electron drift velocity. The word drift means a gradual average movement in one direction, while velocity means speed with direction. Therefore, drift velocity describes both how fast the average directed motion is and the direction in which it occurs.

💡 Definition: Electron drift velocity is the average velocity acquired by free electrons in a conductor due to an applied electric field.

The symbol commonly used for drift velocity is v_d. Its SI unit is the metre per second (m/s).

🔌 2. How Do Electrons Move Inside a Conductor?

Consider a copper wire connected to a battery. Before the battery is connected, free electrons move randomly in different directions. Their individual velocities may be high, but the average velocity of the electrons as a group is approximately zero.

When the battery is connected, an electric field is established in the wire. Since electrons carry negative charge, the electric force on them acts opposite to the direction of the electric field.

The electrons still undergo random motion and frequent collisions. Therefore, they do not travel through the wire in a perfectly straight line. Instead, the electric field produces a small average drift in the direction opposite to the field.

The key distinction is that random electron motion and drift motion are not the same. Drift velocity describes the average directed motion caused by the electric field.

⚡ Direction of electron drift and conventional current

• 🔹 Electric field: Points from higher electric potential to lower electric potential in a simple uniform conductor.
• 🔹 Electron drift: Opposite to the electric field because electrons have negative charge.
• 🔹 Conventional current: In a metal wire, it points in the direction opposite to the electron drift.

Remember that conventional current is defined as the direction in which positive charge would move. This convention is used even when the actual mobile charge carriers in a metal are negatively charged electrons.

📐 3. Derivation of the Electron Drift Velocity Formula

We will derive the relationship between drift velocity and electric current by considering the number of free electrons crossing the cross-section of a conductor.

Step 1: Consider a conductor

Suppose a uniform conductor has the following properties:

• n = number of free electrons per unit volume, measured in m⁻³.
• A = cross-sectional area of the conductor, measured in m².
• v_d = magnitude of electron drift velocity, measured in m/s.
• e = magnitude of the charge of one electron, equal to approximately 1.602 × 10⁻¹⁹ C.
• I = magnitude of electric current, measured in amperes (A).

A cross-section is an imaginary surface cut across the conductor, perpendicular to its length.

Step 2: Calculate the distance travelled

Suppose the electrons have a steady average drift speed v_d. During a time interval Δt, the average drift distance is:

\[ \text{Distance} = v_d\Delta t \]

This follows from the basic relationship:

\[ \text{Distance} = \text{Speed}\times\text{Time} \]

Step 3: Calculate the volume of the conductor considered

Imagine a cylindrical section of the wire whose length is v_dΔt and whose cross-sectional area is A.

The volume of this section is:

\[ \begin{aligned} \text{Volume} &= \text{Area}\times\text{Length}\\ &= A(v_d\Delta t) \end{aligned} \]

Therefore,

\[ \boxed{\text{Volume}=Av_d\Delta t} \]

Step 4: Calculate the number of electrons

The number density n tells us how many free electrons are present in one cubic metre of the conductor.

Thus:

\[ \text{Number of electrons} = n\times\text{Volume} \] \[ \boxed{N=nAv_d\Delta t} \]

Step 5: Calculate the total charge crossing the cross-section

The magnitude of the charge carried by each electron is e. If N electrons cross the cross-section, the magnitude of the total charge transported is:

\[ Q=Ne \]

Substituting N = nAv_dΔt:

\[ Q=(nAv_d\Delta t)e \] \[ \boxed{Q=neAv_d\Delta t} \]

Step 6: Use the definition of electric current

Electric current is the rate at which electric charge crosses a cross-section. Therefore:

\[ I=\frac{Q}{\Delta t} \]

Substitute Q = neAv_dΔt:

\[ I=\frac{neAv_d\Delta t}{\Delta t} \]

Cancel Δt from the numerator and denominator:

\[ \boxed{I=neAv_d} \]

This is the fundamental relationship between electric current and electron drift velocity for a uniform conductor with one dominant type of charge carrier.

Step 7: Obtain the formula for drift velocity

Starting from:

\[ I=neAv_d \]

Divide both sides by neA:

\[ \boxed{v_d=\frac{I}{neA}} \]

✅ Final formula: The magnitude of electron drift velocity is equal to electric current divided by the product of electron number density, electron charge magnitude, and conductor cross-sectional area.

📊 4. Meaning of Each Symbol in the Formula

Symbol	Meaning	SI unit
I	Electric current	Ampere (A)
n	Free-electron number density	m−3
e	Magnitude of electron charge	Coulomb (C)
A	Cross-sectional area	m2
vd	Drift velocity magnitude	m/s
Q	Charge transported	Coulomb (C)

📏 5. Relationship Between Current Density and Drift Velocity

Current density describes how much electric current passes through a unit area perpendicular to the flow of charge. Think of it as current per unit area: the same current spread through a larger area gives a smaller current density. It is represented by J and its SI unit is ampere per square metre (A/m²).

For a uniform current distributed over a cross-sectional area A, the magnitude of current density is:

\[ J=\frac{I}{A} \]

From the drift-velocity relationship:

\[ I=neAv_d \]

Divide both sides by A:

\[ \frac{I}{A}=nev_d \]

Since J = I/A, we obtain:

\[ \boxed{J=nev_d} \]

This equation gives the magnitude of current density for electrons drifting in one direction. In vector form, when the electron drift velocity vector is used, the conventional current density points in the opposite direction:

\[ \boxed{\vec J=-ne\vec v_d} \]

The negative sign appears because the charge of an electron is −e. Here, e denotes the positive magnitude of electron charge.

General relationship between current and current density

If current density varies over the cross-section or is not perpendicular to the area everywhere, current is calculated using the surface integral:

\[ \boxed{I=\int_S \vec J\cdot d\vec A} \]

Here, dA is a small area vector normal to the surface S, and the dot product selects the component of current density passing through that surface.

For uniform current density perpendicular to the cross-section, this simplifies to I = JA.

🧮 6. Solved Numerical Example 1: Calculate Drift Velocity

Question: A copper conductor carries a current density of 1.55 × 10⁶ A/m². Assume the free-electron number density is 8.5 × 10²⁸ m⁻³ and the electron charge magnitude is 1.602 × 10⁻¹⁹ C. Calculate the magnitude of electron drift velocity.

Given data:

• J = 1.55 × 10⁶ A/m²
• n = 8.5 × 10²⁸ m⁻³
• e = 1.602 × 10⁻¹⁹ C

Step 1: Select the formula.

\[ J=nev_d \]

Step 2: Rearrange the formula.

\[ v_d=\frac{J}{ne} \]

Step 3: Substitute the values.

\[ v_d= \frac{1.55\times10^6} {(8.5\times10^{28})(1.602\times10^{-19})} \]

Step 4: Simplify.

First, multiply the values in the denominator:

\[ 8.5\times1.602=13.617 \] \[ 10^{28}\times10^{-19}=10^9 \]

Therefore:

\[ v_d=\frac{1.55\times10^6}{13.617\times10^9} \] \[ v_d\approx1.138\times10^{-4}\ \text{m/s} \]

Convert metres per second into centimetres per second. Since 1 m = 100 cm:

\[ v_d\approx0.01138\ \text{cm/s} \]

✅ Answer: The magnitude of electron drift velocity is approximately 1.14 × 10⁻⁴ m/s.

💡 Learning point: Drift velocity in a metal can be very small even when a measurable current flows through the conductor.

🧮 7. Solved Numerical Example 2: Calculate Electric Current

Question: A conductor has a free-electron number density of 1 × 10²⁹ electrons/m³. Its cross-sectional area is 1 cm², and the magnitude of electron drift velocity is 1.5 × 10⁻⁴ m/s. Calculate the magnitude of electric current. Take e = 1.6 × 10⁻¹⁹ C.

Given data:

• n = 1 × 10²⁹ m⁻³
• A = 1 cm²
• v_d = 1.5 × 10⁻⁴ m/s
• e = 1.6 × 10⁻¹⁹ C

Step 1: Convert the area into square metres.

Since 1 cm = 10⁻² m, square both sides:

\[ 1\ \text{cm}^2=10^{-4}\ \text{m}^2 \]

Step 2: Select the formula.

\[ I=neAv_d \]

Step 3: Substitute the values.

\[ I=(10^{29})(1.6\times10^{-19}) (10^{-4})(1.5\times10^{-4}) \]

Step 4: Multiply the numerical values.

\[ 1.6\times1.5=2.4 \]

Combine the powers of ten:

\[ 10^{29}\times10^{-19} \times10^{-4}\times10^{-4}=10^2 \] \[ I=2.4\times10^2 \] \[ \boxed{I=240\ \text{A}} \]

✅ Answer: The magnitude of electric current is 240 A.

🌩️ 8. Drift Velocity in Terms of Electric Field

Drift velocity can also be related to the electric field inside a conductor.

When an electron is placed in an electric field E, the force on it is:

\[ \vec F=q\vec E \]

For an electron, q = −e. Therefore:

\[ \vec F=-e\vec E \]

According to Newton's second law, force equals mass multiplied by acceleration:

\[ \vec F=m\vec a \]

For an electron of mass m, the instantaneous acceleration between collisions is:

\[ \vec a=-\frac{e}{m}\vec E \]

In the classical Drude model of electrical conduction, electrons undergo frequent collisions. Let τ represent the mean relaxation time, which is the average time between momentum-randomising collisions in this model.

The average drift velocity predicted by this model is:

\[ \boxed{\vec v_d=-\frac{e\tau}{m}\vec E} \]

The magnitude is:

\[ \boxed{v_d=\frac{e\tau}{m}E} \]

This classical formula assumes a simple metal-conduction model and a uniform electric field. Real materials may require more advanced models for precise predictions.

🔹 Electron Mobility

Electron mobility describes how readily electrons acquire drift velocity under an electric field. It is represented by μ.

For the classical model:

\[ \mu=\frac{e\tau}{m} \]

Therefore:

\[ \boxed{\vec v_d=-\mu\vec E} \]

In magnitude:

\[ \boxed{v_d=\mu E} \]

The SI unit of mobility is m²/(V·s).

🔗 9. Relationship Between Current Density, Conductivity and Electric Field

For electron conduction, the vector current-density relationship is:

\[ \vec J=-ne\vec v_d \]

Using the classical drift formula:

\[ \vec v_d=-\frac{e\tau}{m}\vec E \]

Substitute it into the current-density equation:

\[ \vec J=-ne\left(-\frac{e\tau}{m}\vec E\right) \] \[ \vec J=\frac{ne^2\tau}{m}\vec E \]

For an ohmic conductor under these assumptions, current density is proportional to electric field:

\[ \boxed{\vec J=\sigma\vec E} \]

Here, σ is electrical conductivity. Comparing the two expressions gives:

\[ \boxed{\sigma=\frac{ne^2\tau}{m}=ne\mu} \]

Electrical conductivity measures how readily a material allows electric current to flow. Its symbol is σ and its SI unit is siemens per metre (S/m). Do not confuse conductivity with conductance: conductance (G), measured in siemens (S), describes an entire component and depends on its dimensions as well as its material. For a uniform conductor, G = σA/L, where A is cross-sectional area and L is length.

The reciprocal of conductivity is resistivity, represented by ρ. Resistivity describes how strongly a material opposes current, while resistance describes the opposition of a particular component. For a uniform conductor, resistance is R = ρL/A and conductance is G = 1/R:

\[ \boxed{\rho=\frac{1}{\sigma}} \]

The SI unit of resistivity is ohm-metre (Ω·m).

⚖️ 10. Factors Affecting Electron Drift Velocity

For a uniform conductor carrying a given current, drift velocity is determined by:

\[ v_d=\frac{I}{neA} \]

• 🔹 Electric current (I): If all other quantities remain constant, greater current means greater drift-velocity magnitude.
• 🔹 Electron number density (n): For a fixed current and area, a larger number density means a smaller drift-velocity magnitude.
• 🔹 Cross-sectional area (A): For a fixed current and material, a larger cross-sectional area means a smaller drift-velocity magnitude.
• 🔹 Electric field (E): In the classical linear-response model, drift-velocity magnitude is proportional to electric-field magnitude.
• 🔹 Material properties: Mobility and scattering processes affect the drift response to an electric field.

🆚 11. Difference Between Drift Velocity and Random Velocity

Drift velocity	Random electron velocity
Average directed motion due to an electric field.	Irregular motion associated with the electrons' thermal energy and microscopic state.
Produces a net electron flow in a conducting metal.	Random motion alone produces no sustained net current at equilibrium.
Usually much smaller than the characteristic microscopic electron speeds in a metal.	Can be much larger than the drift speed.
Its direction is opposite to the electric field for electrons.	Its direction varies from electron to electron and over time.

🚀 12. Does Electric Current Travel at the Drift Velocity?

No. This is a common misconception.

The drift velocity is the average directed velocity of the charge carriers. When a circuit is switched on, the electromagnetic signal that establishes the electric field propagates through the circuit at a speed determined by the electromagnetic properties and geometry of the system. It is not equal to the slow drift speed of individual electrons.

Therefore, a lamp can respond soon after a switch is closed even though the average drift velocity of electrons in the metal wire is very small.

📝 13. Important Formula Summary

Quantity	Formula
Electron drift velocity from current	\(v_d=\frac{I}{neA}\)
Electric current magnitude	\(I=neAv_d\)
Current density magnitude	\(J=\frac{I}{A}\)
Current density magnitude for electron drift	\(J=nev_d\)
Vector current density for electrons	\(\vec J=-ne\vec v_d\)
Classical drift velocity in an electric field	\(\vec v_d=-\frac{e\tau}{m}\vec E\)
Drift velocity using mobility	\(\vec v_d=-\mu\vec E\)
Electrical conductivity in the classical model	\(\sigma=\frac{ne^2\tau}{m}=ne\mu\)

🎯 14. Remember Points for Exams

✅ Electron drift velocity is the average directed velocity acquired by free electrons under an applied electric field.

✅ The SI unit of drift velocity is m/s.

✅ Electrons drift opposite to the electric field.

✅ In a metal conductor, conventional current points opposite to electron drift.

✅ The current magnitude is given by I = neAv_d.

✅ Drift-velocity magnitude is given by v_d = I/(neA).

✅ Current density has the SI unit A/m².

✅ For uniform electron drift, the current-density magnitude is J = nev_d.

✅ The vector equation for electron current density is J⃗ = −ne v⃗_d.

✅ Drift velocity is generally much smaller than the microscopic random speeds of electrons in a metal.

✅ The electromagnetic signal establishing a circuit's electric field does not travel at the electron drift velocity.

✅ Always convert area into m² and use consistent SI units before substituting values into formulas.

⚡ 50 Important Objective Questions on Electron Drift Velocity — Detailed Solutions

⚡ Core formulas to remember
Current: \(I=\frac{Q}{\Delta t}\)
Current due to electron drift: \(I=neAv_d\)
Drift velocity magnitude: \(v_d=\frac{I}{neA}\)
Current density magnitude: \(J=\frac{I}{A}=nev_d\)

📑 Contents

1. Definition and basic concepts
2. Drift velocity formula and symbols
3. Current density and direction
4. Electric field, mobility and conductivity
5. Numerical problems and applications
6. Conductance and final revision

⚡ MCQ 1: What is electron drift velocity?

Options:

• A) The maximum speed of an electron in a metal
• B) The average directed velocity of free electrons caused by an applied electric field
• C) The speed of an electromagnetic signal in a wire
• D) The random thermal velocity of one electron

✅ Correct Answer: B) The average directed velocity of free electrons caused by an applied electric field

🧠 Detailed Explanation:

Electron drift velocity is the average directed velocity acquired by free electrons when an electric field is applied. Individual electrons continue to move and scatter, so drift velocity describes the average motion of the electron population rather than the full instantaneous velocity of one electron.

📌 Remember Point: Drift velocity is an average directed motion; its SI unit is m/s.

⚡ MCQ 2: What is the SI unit of drift velocity?

Options:

• A) m/s
• B) A/m²
• C) C/s
• D) Ω·m

✅ Correct Answer: A) m/s

🧠 Detailed Explanation:

Velocity is the rate of change of position with time. Its SI unit is metre per second (m/s). Current density is measured in A/m², current in amperes (equivalent to C/s), and resistivity in Ω·m.

📌 Remember Point: Always distinguish velocity (m/s) from current density (A/m²).

⚡ MCQ 3: Which particles mainly carry electric current in an ordinary metallic wire?

Options:

• A) Protons moving through the wire
• B) Neutrons moving through the wire
• C) Free electrons
• D) The nuclei of copper atoms flowing along the wire

✅ Correct Answer: C) Free electrons

🧠 Detailed Explanation:

Metals contain mobile electrons that can move through the material. The positively charged atomic nuclei remain bound in the metal lattice and do not travel along the wire as the charge carriers in ordinary conduction.

📌 Remember Point: In metallic conductors, free electrons are the mobile charge carriers.

⚡ MCQ 4: What happens to free electrons in a metal before an external electric field is applied?

Options:

• A) They are all stationary
• B) They move randomly, with approximately zero average drift at equilibrium
• C) They all move in the same direction
• D) They move only from negative to positive terminal

✅ Correct Answer: B) They move randomly, with approximately zero average drift at equilibrium

🧠 Detailed Explanation:

Free electrons have microscopic motion even without an applied field. At equilibrium, their random motions cancel statistically, so there is no sustained net drift current through the conductor.

📌 Remember Point: Random motion can exist without a net current.

⚡ MCQ 5: When an electric field is applied to a metallic conductor, electron drift is generally:

Options:

• A) In the same direction as the electric field
• B) Perpendicular to the electric field
• C) Opposite to the electric field
• D) Always vertically downward

✅ Correct Answer: C) Opposite to the electric field

🧠 Detailed Explanation:

The force on a charge is given by the vector equation F = qE. An electron has charge q = −e, so its electric force is opposite to the electric field. Its average drift is therefore opposite to the field in an ordinary metal.

📌 Remember Point: Negative charge carriers drift opposite to the electric field.

⚡ MCQ 6: In a metallic conductor, conventional current points:

Options:

• A) In the same direction as electron drift
• B) Opposite to electron drift
• C) Always opposite to the electric field in every material
• D) In a random direction

✅ Correct Answer: B) Opposite to electron drift

🧠 Detailed Explanation:

Conventional current is defined as the direction positive charge would move. Because electrons are negatively charged, their drift direction is opposite to conventional current in a metal wire.

📌 Remember Point: Conventional current and electron flow have opposite directions in metals.

⚡ MCQ 7: Which equation gives the magnitude of electric current in terms of electron number density and drift velocity?

Options:

• A) I = neAv_d
• B) I = n/(eAv_d)
• C) I = eA/(nv_d)
• D) I = neA/v_d

✅ Correct Answer: A) I = neAv_d

🧠 Detailed Explanation:

In a time Δt, electrons with drift speed v_d move an average distance v_dΔt. The swept volume is Av_dΔt, so the number of electrons is nAv_dΔt. Their total charge magnitude is neAv_dΔt. Dividing by Δt gives I = neAv_d.

📌 Remember Point: Current magnitude: I = neAv_d.

⚡ MCQ 8: Which is the correct formula for the magnitude of electron drift velocity?

Options:

• A) v_d = neA/I
• B) v_d = I/(neA)
• C) v_d = I neA
• D) v_d = ne/(IA)

✅ Correct Answer: B) v_d = I/(neA)

🧠 Detailed Explanation:

Start with I = neAv_d. To isolate v_d, divide both sides by neA. This gives v_d = I/(neA). Use consistent SI units when substituting values.

📌 Remember Point: Drift speed is current divided by n, e, and area.

⚡ MCQ 9: In the drift velocity formula v_d = I/(neA), what does n represent?

Options:

• A) The number of electrons crossing per second
• B) Electron mass
• C) Free-electron number density per unit volume
• D) The number of volts across the conductor

✅ Correct Answer: C) Free-electron number density per unit volume

🧠 Detailed Explanation:

The symbol n represents the number of mobile electrons per cubic metre. Its SI unit is m⁻³. It describes how many charge carriers are available in a unit volume of the material.

📌 Remember Point: Number density n is measured in m⁻³.

⚡ MCQ 10: What is the SI unit of electron number density n?

Options:

• A) m⁻³
• B) m³
• C) C/m
• D) A/m²

✅ Correct Answer: A) m⁻³

🧠 Detailed Explanation:

Number density means the number of particles per unit volume. Since volume is measured in cubic metres, its unit is m⁻³. The particle count itself is dimensionless.

📌 Remember Point: Number per unit volume has units of inverse volume.

⚡ MCQ 11: What is the magnitude of the charge of one electron, approximately?

Options:

• A) 1.602 × 10⁻¹⁹ C
• B) 1.602 × 10¹⁹ C
• C) 9.11 × 10⁻³¹ C
• D) 6.626 × 10⁻³⁴ C

✅ Correct Answer: A) 1.602 × 10⁻¹⁹ C

🧠 Detailed Explanation:

The electron's actual charge is −1.602 × 10⁻¹⁹ C. In formulas for magnitudes, e usually denotes the positive magnitude 1.602 × 10⁻¹⁹ C. The value 9.11 × 10⁻³¹ is approximately the electron mass in kilograms.

📌 Remember Point: Electron charge q_e = −e; charge magnitude e ≈ 1.602 × 10⁻¹⁹ C.

⚡ MCQ 12: In I = neAv_d, what does A represent?

Options:

• A) Electric potential
• B) Cross-sectional area of the conductor
• C) Acceleration of the electron
• D) Electrical conductivity

✅ Correct Answer: B) Cross-sectional area of the conductor

🧠 Detailed Explanation:

A is the cross-sectional area perpendicular to the conductor's length. It must be expressed in square metres (m²) when using SI values in the formula.

📌 Remember Point: Convert mm² or cm² to m² before calculation.

⚡ MCQ 13: A conductor has cross-sectional area 2 mm². What is this area in m²?

Options:

• A) 2 × 10⁻³ m²
• B) 2 × 10⁻⁶ m²
• C) 2 × 10⁻⁴ m²
• D) 2 × 10⁶ m²

✅ Correct Answer: B) 2 × 10⁻⁶ m²

🧠 Detailed Explanation:

Since 1 mm = 10⁻³ m, squaring gives 1 mm² = 10⁻⁶ m². Therefore, 2 mm² = 2 × 10⁻⁶ m².

📌 Remember Point: Area conversions require squaring the length conversion factor.

⚡ MCQ 14: What is electric current, by definition?

Options:

• A) Charge multiplied by time
• B) Charge passing per unit time
• C) Energy per unit volume
• D) Resistance per unit area

✅ Correct Answer: B) Charge passing per unit time

🧠 Detailed Explanation:

Electric current is the rate at which electric charge crosses a chosen surface. For steady current, I = Q/Δt. One ampere is one coulomb per second.

📌 Remember Point: I = Q/t for steady current; 1 A = 1 C/s.

⚡ MCQ 15: A charge of 12 C passes through a wire in 3 s. What is the current?

Options:

• A) 4 A
• B) 9 A
• C) 15 A
• D) 36 A

✅ Correct Answer: A) 4 A

🧠 Detailed Explanation:

Use I = Q/t. Substituting Q = 12 C and t = 3 s gives I = 12/3 = 4 A.

📌 Remember Point: For steady current, divide charge by time.

⚡ MCQ 16: A wire carries 4 A. If all other quantities remain constant, what happens to drift speed when current doubles?

Options:

• A) It becomes half
• B) It remains unchanged
• C) It doubles
• D) It becomes four times

✅ Correct Answer: C) It doubles

🧠 Detailed Explanation:

From v_d = I/(neA), drift speed is directly proportional to current when n, e, and A are fixed. Doubling I therefore doubles v_d.

📌 Remember Point: At fixed material and area, v_d ∝ I.

⚡ MCQ 17: For fixed current, electron density, and charge, if the conductor's cross-sectional area doubles, the drift speed:

Options:

• A) Doubles
• B) Becomes half
• C) Becomes four times
• D) Does not change

✅ Correct Answer: B) Becomes half

🧠 Detailed Explanation:

The formula v_d = I/(neA) shows that drift speed is inversely proportional to area when current and material properties remain fixed. Doubling A makes the drift speed half as large.

📌 Remember Point: At fixed current and material, v_d ∝ 1/A.

⚡ MCQ 18: For a fixed current and cross-sectional area, if the number density n doubles, drift speed:

Options:

• A) Doubles
• B) Becomes half
• C) Becomes four times
• D) Does not change

✅ Correct Answer: B) Becomes half

🧠 Detailed Explanation:

Because v_d = I/(neA), drift speed is inversely proportional to n when the other quantities remain constant. Twice as many available carriers can carry the same current with half the average drift speed.

📌 Remember Point: At fixed I and A, v_d ∝ 1/n.

⚡ MCQ 19: What is current density?

Options:

• A) Current per unit cross-sectional area perpendicular to current
• B) Charge multiplied by resistance
• C) Voltage per unit time
• D) The number of atoms in a wire

✅ Correct Answer: A) Current per unit cross-sectional area perpendicular to current

🧠 Detailed Explanation:

Current density describes current distributed over an area. For uniform current perpendicular to the cross-section, its magnitude is J = I/A and its SI unit is A/m².

📌 Remember Point: J = I/A for uniform perpendicular current density.

⚡ MCQ 20: What is the SI unit of current density?

Options:

• A) A/m²
• B) A·m²
• C) C/s²
• D) Ω/m

✅ Correct Answer: A) A/m²

🧠 Detailed Explanation:

Current density is current divided by area. Since current is measured in amperes and area in square metres, the unit is A/m².

📌 Remember Point: Current density is measured in amperes per square metre.

⚡ MCQ 21: For uniform electron drift, what is the magnitude relationship between current density and drift velocity?

Options:

• A) J = v_d/(ne)
• B) J = nev_d
• C) J = ne/v_d
• D) J = nAv_d

✅ Correct Answer: B) J = nev_d

🧠 Detailed Explanation:

Divide I = neAv_d by A to get I/A = nev_d. Since J = I/A for uniform perpendicular current, J = nev_d for magnitudes.

📌 Remember Point: Magnitude: J = nev_d; vector form: J⃗ = −ne v⃗_d.

⚡ MCQ 22: Why is there a negative sign in the vector equation J⃗ = −ne v⃗_d?

Options:

• A) Current density and electron drift point in opposite directions
• B) Electron mass is negative
• C) Area is negative
• D) Electric current is always negative

✅ Correct Answer: A) Current density and electron drift point in opposite directions

🧠 Detailed Explanation:

The electron charge is −e. Current density is charge density multiplied by carrier velocity, so the negative carrier charge makes conventional current density point opposite to the electron drift velocity.

📌 Remember Point: The minus sign represents the negative charge of electrons.

⚡ MCQ 23: What is the force on an electron in an electric field E?

Options:

• A) F⃗ = eE⃗
• B) F⃗ = −eE⃗
• C) F⃗ = mE⃗
• D) F⃗ = E⃗/e

✅ Correct Answer: B) F⃗ = −eE⃗

🧠 Detailed Explanation:

Electric force is F⃗ = qE⃗. For an electron q = −e, giving F⃗ = −eE⃗. The negative sign indicates the force is opposite to the electric field.

📌 Remember Point: Use F⃗ = qE⃗ and q_e = −e.

⚡ MCQ 24: What is the SI unit of electric field?

Options:

• A) V/m
• B) A/m²
• C) C·m
• D) Ω·m

✅ Correct Answer: A) V/m

🧠 Detailed Explanation:

Electric field can be expressed as force per unit charge (N/C), which is equivalent to volts per metre (V/m).

📌 Remember Point: Electric field units: N/C or V/m.

⚡ MCQ 25: According to Newton's second law, force is equal to:

Options:

• A) Mass divided by acceleration
• B) Mass multiplied by acceleration
• C) Velocity multiplied by charge
• D) Current multiplied by area

✅ Correct Answer: B) Mass multiplied by acceleration

🧠 Detailed Explanation:

Newton's second law gives F⃗ = m a⃗ for constant mass. Combining this with the electric force on an electron gives its instantaneous acceleration between collisions: a⃗ = −(e/m)E⃗.

📌 Remember Point: F⃗ = m a⃗.

⚡ MCQ 26: In the classical Drude model, what does the mean relaxation time τ describe?

Options:

• A) The time for light to cross a wire
• B) A characteristic average time scale for momentum relaxation due to scattering
• C) The lifetime of an electron
• D) The time needed to charge a capacitor in every circuit

✅ Correct Answer: B) A characteristic average time scale for momentum relaxation due to scattering

🧠 Detailed Explanation:

The Drude model uses τ to represent the average time scale over which electron momentum is relaxed by scattering in the material. It is used to relate the electric field to average drift velocity.

📌 Remember Point: In the Drude model, τ is a momentum-relaxation time scale.

⚡ MCQ 27: According to the classical Drude model, which vector equation gives electron drift velocity in a uniform field?

Options:

• A) v⃗_d = (eτ/m)E⃗
• B) v⃗_d = −(eτ/m)E⃗
• C) v⃗_d = mE⃗/(eτ)
• D) v⃗_d = τE⃗/(em)

✅ Correct Answer: B) v⃗_d = −(eτ/m)E⃗

🧠 Detailed Explanation:

The classical model predicts v⃗_d = −(eτ/m)E⃗. The minus sign indicates electron drift opposite to the field. The magnitude is v_d = eτE/m.

📌 Remember Point: Classical model: v⃗_d = −(eτ/m)E⃗.

⚡ MCQ 28: What is electron mobility?

Options:

• A) Current divided by area
• B) Drift-velocity magnitude per unit electric-field magnitude in the linear model
• C) Electron charge multiplied by mass
• D) Resistance per unit length

✅ Correct Answer: B) Drift-velocity magnitude per unit electric-field magnitude in the linear model

🧠 Detailed Explanation:

Mobility μ describes how much drift velocity develops per unit electric field in the linear-response regime. Its SI unit is m²/(V·s). In the classical Drude model, μ = eτ/m.

📌 Remember Point: Magnitude relationship: v_d = μE.

⚡ MCQ 29: What is the SI unit of mobility?

Options:

• A) m²/(V·s)
• B) V/m
• C) A/m²
• D) Ω·m

✅ Correct Answer: A) m²/(V·s)

🧠 Detailed Explanation:

Mobility is drift speed divided by electric-field magnitude. Its units are (m/s)/(V/m) = m²/(V·s).

📌 Remember Point: Mobility unit: m² per volt-second.

⚡ MCQ 30: If mobility remains constant and electric-field magnitude doubles, the drift-speed magnitude in the linear-response model:

Options:

• A) Halves
• B) Doubles
• C) Remains unchanged
• D) Becomes zero

✅ Correct Answer: B) Doubles

🧠 Detailed Explanation:

The magnitude relation is v_d = μE. If μ is constant, drift speed is directly proportional to E. Doubling the field doubles the drift-speed magnitude within the model's linear-response range.

📌 Remember Point: For constant mobility, v_d ∝ E.

⚡ MCQ 31: Which equation relates current density and electric field for an ohmic material in the linear regime?

Options:

• A) J⃗ = σE⃗
• B) J⃗ = ρE⃗²
• C) J⃗ = E⃗/σ²
• D) J⃗ = σ/E⃗

✅ Correct Answer: A) J⃗ = σE⃗

🧠 Detailed Explanation:

The microscopic form of Ohm's law is J⃗ = σE⃗, where σ is conductivity. This linear relationship applies when the material behaves ohmically under the stated conditions.

📌 Remember Point: Microscopic Ohm's law: J⃗ = σE⃗.

⚡ MCQ 32: What is the SI unit of electrical conductivity σ?

Options:

• A) S/m
• B) S
• C) Ω·m
• D) m/s

✅ Correct Answer: A) S/m

🧠 Detailed Explanation:

Conductivity is a material property that describes how readily a material conducts current. Its SI unit is siemens per metre (S/m). Conductance of a component, by contrast, is measured in siemens (S).

📌 Remember Point: Conductivity: S/m; conductance: S.

⚡ MCQ 33: Which formula relates conductivity and mobility for electron conduction in the simple model?

Options:

• A) σ = neμ
• B) σ = n/(eμ)
• C) σ = eμ/n
• D) σ = n + e + μ

✅ Correct Answer: A) σ = neμ

🧠 Detailed Explanation:

From J⃗ = −ne v⃗_d and v⃗_d = −μE⃗, substitute to obtain J⃗ = neμE⃗. Comparing with J⃗ = σE⃗ gives σ = neμ.

📌 Remember Point: For the simple electron model, σ = neμ.

⚡ MCQ 34: In the classical Drude model, conductivity is:

Options:

• A) σ = ne²τ/m
• B) σ = m/(ne²τ)
• C) σ = nme/τ
• D) σ = eτ/(nm)

✅ Correct Answer: A) σ = ne²τ/m

🧠 Detailed Explanation:

The classical model gives μ = eτ/m. Substituting this into σ = neμ gives σ = ne(eτ/m) = ne²τ/m.

📌 Remember Point: Drude conductivity: σ = ne²τ/m.

⚡ MCQ 35: What is the relationship between resistivity ρ and conductivity σ?

Options:

• A) ρ = σ
• B) ρ = 1/σ
• C) ρ = σ²
• D) ρ = 1 + σ

✅ Correct Answer: B) ρ = 1/σ

🧠 Detailed Explanation:

Resistivity and conductivity are reciprocals for an isotropic material in the usual linear model. A highly conductive material has low resistivity. Conductivity is measured in S/m and resistivity in Ω·m.

📌 Remember Point: ρ = 1/σ.

⚡ MCQ 36: Which statement correctly distinguishes conductance from conductivity?

Options:

• A) They are always identical quantities
• B) Conductance is measured in S and depends on component geometry; conductivity is a material property measured in S/m
• C) Conductance is measured in m/s
• D) Conductivity depends only on wire length

✅ Correct Answer: B) Conductance is measured in S and depends on component geometry; conductivity is a material property measured in S/m

🧠 Detailed Explanation:

Conductance G describes how easily a particular component passes current and is measured in siemens (S). Conductivity σ describes the material and is measured in S/m. For a uniform conductor, G = σA/L.

📌 Remember Point: For a uniform conductor, G = σA/L and G = 1/R.

⚡ MCQ 37: For a uniform conductor of length L and area A, which equation gives its conductance using conductivity?

Options:

• A) G = σL/A
• B) G = σA/L
• C) G = ρA/L
• D) G = L/(σA)

✅ Correct Answer: B) G = σA/L

🧠 Detailed Explanation:

For a uniform conductor, resistance is R = ρL/A. Since σ = 1/ρ, this can be written R = L/(σA). Conductance is G = 1/R, so G = σA/L.

📌 Remember Point: Conductance depends on material conductivity and geometry.

⚡ MCQ 38: A uniform wire has conductivity 5 S/m, area 2 m², and length 10 m. What is its conductance?

Options:

• A) 1 S
• B) 10 S
• C) 25 S
• D) 0.1 S

✅ Correct Answer: A) 1 S

🧠 Detailed Explanation:

Use G = σA/L. Substituting gives G = (5 × 2)/10 = 1 S. This example uses simplified dimensions to practise the formula.

📌 Remember Point: G = σA/L; conductance is measured in siemens.

⚡ MCQ 39: What is the SI unit of resistivity?

Options:

• A) Ω·m
• B) S/m
• C) Ω/m²
• D) A/m

✅ Correct Answer: A) Ω·m

🧠 Detailed Explanation:

Resistivity ρ is a material property and its SI unit is ohm-metre (Ω·m). For a uniform conductor, R = ρL/A.

📌 Remember Point: Resistivity: Ω·m; resistance: Ω.

⚡ MCQ 40: A current density is 1.55 × 10⁶ A/m², n = 8.5 × 10²⁸ m⁻³, and e = 1.602 × 10⁻¹⁹ C. What is the approximate drift-speed magnitude?

Options:

• A) 1.14 × 10⁻⁴ m/s
• B) 1.14 × 10⁴ m/s
• C) 1.14 × 10⁻² m/s
• D) 1.14 m/s

✅ Correct Answer: A) 1.14 × 10⁻⁴ m/s

🧠 Detailed Explanation:

Use J = nev_d, so v_d = J/(ne). The denominator is (8.5 × 1.602) × 10^(28−19) = 13.617 × 10⁹. Thus v_d = (1.55 × 10⁶)/(13.617 × 10⁹) ≈ 1.14 × 10⁻⁴ m/s.

📌 Remember Point: For current density data, use v_d = J/(ne).

⚡ MCQ 41: A wire carries 5 A, has area 1 mm², and n = 8.5 × 10²⁸ m⁻³. Taking e = 1.602 × 10⁻¹⁹ C, what is the approximate drift speed?

Options:

• A) 3.67 × 10⁻⁴ m/s
• B) 3.67 × 10⁴ m/s
• C) 3.67 × 10⁻⁶ m/s
• D) 1.36 × 10⁻³ m/s

✅ Correct Answer: A) 3.67 × 10⁻⁴ m/s

🧠 Detailed Explanation:

Convert the area: 1 mm² = 10⁻⁶ m². Then v_d = I/(neA) = 5/[(8.5 × 10²⁸)(1.602 × 10⁻¹⁹)(10⁻⁶)]. The denominator is 13.617 × 10³, giving v_d ≈ 3.67 × 10⁻⁴ m/s.

📌 Remember Point: Convert area to m² before using v_d = I/(neA).

⚡ MCQ 42: A conductor has n = 10²⁹ m⁻³, A = 1 cm², v_d = 1.5 × 10⁻⁴ m/s, and e = 1.6 × 10⁻¹⁹ C. What current flows?

Options:

• A) 24 A
• B) 240 A
• C) 2.4 A
• D) 2400 A

✅ Correct Answer: B) 240 A

🧠 Detailed Explanation:

Convert A = 1 cm² = 10⁻⁴ m². Use I = neAv_d = (10²⁹)(1.6 × 10⁻¹⁹)(10⁻⁴)(1.5 × 10⁻⁴). The numerical product is 2.4 and the combined power is 10², so I = 240 A.

📌 Remember Point: Use I = neAv_d and convert cm² to m².

⚡ MCQ 43: If 6 × 10¹⁸ electrons pass a cross-section each second, what is the current magnitude? Use e = 1.6 × 10⁻¹⁹ C.

Options:

• A) 0.96 A
• B) 9.6 A
• C) 96 A
• D) 0.096 A

✅ Correct Answer: A) 0.96 A

🧠 Detailed Explanation:

Each second, the transported charge magnitude is N e = (6 × 10¹⁸)(1.6 × 10⁻¹⁹) C = 0.96 C. Since current is charge per second, I = 0.96 A.

📌 Remember Point: Current = charge per second; multiply carrier rate by charge per carrier.

⚡ MCQ 44: A uniform conductor carries 10 A through an area of 2 × 10⁻⁴ m². What is the current-density magnitude?

Options:

• A) 5 × 10⁴ A/m²
• B) 2 × 10⁻³ A/m²
• C) 5 × 10⁻⁵ A/m²
• D) 2 × 10⁵ A/m²

✅ Correct Answer: A) 5 × 10⁴ A/m²

🧠 Detailed Explanation:

For uniform perpendicular current, J = I/A = 10/(2 × 10⁻⁴) = 5 × 10⁴ A/m².

📌 Remember Point: Current density equals current divided by area.

⚡ MCQ 45: If the current stays the same while a wire's area becomes three times larger, what happens to current density?

Options:

• A) It becomes three times larger
• B) It becomes one-third as large
• C) It remains the same
• D) It becomes nine times larger

✅ Correct Answer: B) It becomes one-third as large

🧠 Detailed Explanation:

For uniform current, J = I/A. With I fixed, current density is inversely proportional to area. Tripling area reduces current density to one-third.

📌 Remember Point: At fixed current, J ∝ 1/A.

⚡ MCQ 46: A uniform conductor has a current density of 2 × 10⁵ A/m² and area 3 × 10⁻⁶ m². What is the current?

Options:

• A) 0.6 A
• B) 6 A
• C) 60 A
• D) 6 × 10¹¹ A

✅ Correct Answer: A) 0.6 A

🧠 Detailed Explanation:

Use I = JA. Therefore I = (2 × 10⁵)(3 × 10⁻⁶) = 6 × 10⁻¹ = 0.6 A.

📌 Remember Point: For uniform perpendicular current density, I = JA.

⚡ MCQ 47: What is the volume swept by electrons' average drift in time Δt through a wire of area A?

Options:

• A) A/(v_dΔt)
• B) Av_dΔt
• C) A + v_d + Δt
• D) neAv_d

✅ Correct Answer: B) Av_dΔt

🧠 Detailed Explanation:

The average drift distance during Δt is v_dΔt. The volume of the corresponding wire section is area multiplied by length: V = A(v_dΔt).

📌 Remember Point: Swept volume = cross-sectional area × drift distance.

⚡ MCQ 48: In the derivation of I = neAv_d, what does N = nAv_dΔt represent?

Options:

• A) The resistance of the conductor
• B) The number of electrons in the swept volume
• C) The electric field strength
• D) The current density vector

✅ Correct Answer: B) The number of electrons in the swept volume

🧠 Detailed Explanation:

The swept volume is Av_dΔt. Multiplying this volume by number density n (electrons per cubic metre) gives N = nAv_dΔt electrons. Multiplying N by e gives the charge magnitude crossing in that time.

📌 Remember Point: Number of carriers = number density × volume.

⚡ MCQ 49: Why is the time interval Δt cancelled when deriving I = neAv_d?

Options:

• A) Because time has no unit
• B) Because charge is divided by the same time interval used to calculate the swept volume
• C) Because current is independent of time in every situation
• D) Because electron charge equals time

✅ Correct Answer: B) Because charge is divided by the same time interval used to calculate the swept volume

🧠 Detailed Explanation:

The charge transported in time Δt is Q = neAv_dΔt. For steady current, I = Q/Δt. Substituting Q causes the same Δt factor to cancel, giving I = neAv_d.

📌 Remember Point: The derivation assumes steady average drift during the chosen interval.

⚡ MCQ 50: Which statement about electron drift speed in a metal is generally correct?

Options:

• A) It is always faster than light
• B) It is often much smaller than microscopic electron speeds
• C) It is always exactly 1 m/s
• D) It equals the signal propagation speed

✅ Correct Answer: B) It is often much smaller than microscopic electron speeds

🧠 Detailed Explanation:

The drift speed is the small average directed motion of electrons. Their microscopic motion can be much faster and random. The electrical signal that establishes the field also propagates at a different speed.

📌 Remember Point: Drift speed is not the same as microscopic electron speed or signal speed.

🎯 Final Formula Revision

• Electric current: \(I=Q/\Delta t\)
• Current from drift velocity: \(I=neAv_d\)
• Drift velocity magnitude: \(v_d=I/(neA)\)
• Current density magnitude: \(J=I/A=nev_d\)
• Vector current density for electrons: \(\vec J=-ne\vec v_d\)
• Force on an electron: \(\vec F=-e\vec E\)
• Classical Drude model: \(\vec v_d=-(e\tau/m)\vec E\)
• Mobility: \(\mu=e\tau/m\), and \(v_d=\mu E\) in magnitude
• Conductivity: \(\sigma=ne\mu=ne^2\tau/m\) in the simple Drude model
• Resistivity: \(\rho=1/\sigma\)
• Conductance: \(G=1/R=\sigma A/L\) for a uniform conductor