**Definition**: Electric current is the rate of flow of electric charge through a cross-section of a conductor. Mathematically, if a net charge ΔQ flows through a cross-section during time interval Δt, the instantaneous current is:
**I = lim(Δt→0) ΔQ/Δt = dQ/dt**
For steady current, **I = Q/t** where Q is the total charge flowing in time t.
**SI Unit**: Ampere (A). One ampere is defined through the magnetic effects of currents (studied in Chapter 4). An ampere is the typical order of magnitude of current in domestic appliances. Lightning carries currents in tens of thousands of amperes (~10,000 A), while nerve signals carry microamperes (~10 μA).
**Key Point for Board Exam**: Current is a scalar quantity (it has magnitude only, no direction), though we speak of direction of current as the direction of positive charge flow (opposite to electron flow in conductors).
---
In solid metallic conductors, atoms are tightly bound together. While individual electrons are bound to their atoms at normal conditions, in bulk metals the electron clouds from different atoms overlap significantly. Some electrons become practically free to move within the material—these are called **conduction electrons** or **free electrons**.
**In the absence of electric field**:
**When electric field E is applied**:
**Important Conceptual Point**: A steady electric field in the conductor requires continuous supply of charges at the ends (terminals). Without this, electrons would quickly neutralize the charges and the field would collapse.
---
**Ohm's Law**: The potential difference V across a conductor is directly proportional to the current I flowing through it, provided physical conditions (temperature, pressure, etc.) remain constant.
**V ∝ I**
**V = IR**
where **R** is the **resistance** of the conductor (SI unit: ohm, Ω).
**Resistance R** depends on:
1. **Material** of the conductor
2. **Dimensions** of the conductor (length and cross-sectional area)
**Dimensional Analysis**:
Consider two identical conductor slabs of length l and area A:
**Combining both relations**:
**R = ρ(l/A)**
where **ρ (rho)** is the **resistivity** of the material (SI unit: Ω·m). Resistivity depends only on the material and temperature, not on dimensions.
From **R = ρl/A**, Ohm's law becomes:
**V = I × ρ(l/A)**
**V/l = I/A × ρ**
**E = ρj**
where:
**Vector Form of Ohm's Law**:
**E = ρj** or **j = σE**
where **σ = 1/ρ** is the **conductivity** (SI unit: S/m or Ω⁻¹·m⁻¹; S = siemens).
**Board-Relevant Points**:
---
When an electric field E is applied, each electron experiences a force **F = -eE** and accelerates with:
**a = F/m = -eE/m**
However, electrons undergo **random collisions** with positive ions of the lattice. Between collisions, an electron accelerates; at collision, it loses direction (not speed). This leads to a **net drift** superposed on random thermal motion.
Consider electron i at time t:
**Vᵢ = vᵢ - (eE/m)tᵢ**
Taking ensemble average over all N electrons:
**Average velocity (drift velocity)**:
**v_d = ⟨Vᵢ⟩ = 0 - (eE/m)τ**
**v_d = -(eτ/m)E**
**Magnitude: |v_d| = eEτ/m**
**Key Insight**: Despite continuous acceleration, electrons achieve a **constant average velocity**. This is because they lose velocity in random collisions, reaching an equilibrium where acceleration between collisions is balanced by randomization at collision.
In time Δt, all free electrons within distance |v_d|Δt to the left of an area A perpendicular to E will cross that area.
**Number of electrons crossing = n × A × |v_d| × Δt**
**Charge crossing = n × e × A × |v_d| × Δt**
**Current I = (charge)/(Δt) = n e A |v_d|**
Substituting |v_d| = eEτ/m:
**I = ne A × (eEτ/m) = (ne²Aτ/m) × E**
**Current density j = I/A = (ne²τ/m) × E**
Comparing **j = (ne²τ/m)E** with **j = σE**:
**σ = ne²τ/m**
**ρ = 1/σ = m/(ne²τ)**
**Board-Important Result**: Resistivity arises from:
**Temperature Dependence of Resistivity**:
At higher temperatures, ions vibrate more vigorously → more frequent collisions → τ decreases → ρ increases (nearly linear for metals in normal range).
**ρ(T) ≈ ρ₀[1 + α(T - T₀)]**
where α is the temperature coefficient of resistivity.
---
**Mobility μ** is the magnitude of drift velocity per unit electric field:
**μ = |v_d|/E**
**SI Unit**: m²/Vs (or cm²/Vs in CGS; 1 m²/Vs = 10⁴ cm²/Vs)
**Mobility is always positive**.
From |v_d| = eEτ/m:
**μ = |v_d|/E = eτ/m**
**Conductivity in terms of mobility**:
**σ = ne²τ/m = neμ**
**Key Point**: Mobility characterizes how easily charge carriers move in response to an electric field. Higher mobility means:
---
Ohm's law **V = IR** is valid for many materials over a wide range of conditions, but **deviations occur** in:
Voltage is no longer proportional to current. Examples:
The V-I relation depends on the **sign** of V (polarity). Reversing voltage does not produce equal and opposite current. Examples:
In some materials (ferromagnetic conductors, superconductors), the V-I relation depends on the **history** of voltage application. The curve traced during increase of V differs from the curve during decrease.
**Ohm's Law Conditions for Validity**:
---
**Example**: A copper wire (cross-sectional area A = 1.0 × 10⁻⁷ m²) carries current I = 1.5 A. Estimate the drift speed.
**Given**:
**Number density of conduction electrons**:
n = (Number of atoms per m³) = (ρ_mass × N_A)/M = (9.0 × 10³ × 6.0 × 10²³)/(63.5) = 8.5 × 10²⁸ m⁻³
**Drift velocity** (from I = neAv_d):
v_d = I/(neA) = 1.5/[(8.5 × 10²⁸) × (1.6 × 10⁻¹⁹) × (1.0 × 10⁻⁷)]
v_d = 1.5/(1.36 × 10³) = **1.1 × 10⁻³ m/s = 1.1 mm/s**
**Physical Insight**:
---
| Quantity | Formula | SI Unit |
|----------|---------|---------|
| Current | I = dQ/dt | A (ampere) |
| Resistance | R = ρl/A | Ω (ohm) |
| Resistivity | ρ = m/(ne²τ) | Ω·m |
| Conductivity | σ = 1/ρ = ne²τ/m | S/m (siemens) |
| Drift velocity | v_d = eEτ/m | m/s |
| Current density | j = I/A = σE | A/m² |
| Mobility | μ = eτ/m = |v_d|/E | m²/(V·s) |
| Ohm's Law (microscopic) | j = σE or E = ρj | — |
| Ohm's Law (macroscopic) | V = IR | — |
**Examination Strategy**: Focus on drift velocity derivation, Ohm's law applications to circuits, and understanding why resistivity increases with temperature. Practice numerical problems on drift velocity and current density calculations.
Q1. Electric current is defined as I = ΔQ/Δt. What does ΔQ represent in this expression?
Answer: B — Current is the net charge (forward minus backward) crossing a cross-section per unit time, as stated in the definition given in the study material.
Q2. A conductor of length 50 cm and cross-sectional area 2 cm² has resistivity ρ = 1.7 × 10⁻⁸ Ω·m. Calculate its resistance.
Answer: B — Using R = ρl/A: R = (1.7 × 10⁻⁸ × 0.50) / (2 × 10⁻⁴) = 8.5 × 10⁻⁹ / 2 × 10⁻⁴ = 4.25 × 10⁻⁵ Ω. (Note: Convert cm to m and cm² to m²; 2 cm² = 2 × 10⁻⁴ m².)
Q3. Why is there no net electric current in a conductor when no external electric field is applied?
Answer: B — Electrons undergo random collisions and thermal motion; equal numbers move in any direction versus the opposite direction, yielding zero net current without an applied field.
Q4. If the length of a uniform conductor is doubled while keeping its material and cross-sectional area constant, what happens to its resistance?
Answer: B — Since R = ρl/A and ρ and A are constant, resistance is directly proportional to length; doubling l doubles R, as shown in equation (3.4) of the study material.
Q5. Assertion (A): The drift velocity of electrons in a conductor is of the order of mm/s. Reason (R): The random thermal velocity of electrons is much larger than drift velocity.
Answer: A — Drift velocity is indeed very small (~mm/s) because electrons undergo frequent collisions, and R correctly explains why: thermal speeds are ~10⁵ m/s, so drift is a small net effect superimposed on rapid random motion.
Q6. Which of the following statements is NOT correct regarding Ohm's law and resistance?
Answer: A — Ohm's law applies only to ohmic materials (where R is constant); non-ohmic materials like diodes and thermistors do not obey V = IR universally.
Q7. A cylindrical conductor has cross-sectional area A and length l. Another conductor is made by combining four identical conductors in series (end-to-end, same alignment). What is the resistance of the new combination?
Answer: C — Series combination: total length = 4l, area remains A. Since R ∝ l, the total resistance = 4 × (ρl/A) = 4R.
Q8. In a conductor with applied electric field E, electrons drift with velocity vd. Which of the following is true?
Answer: B — Drift velocity vd ∝ E-field (stronger field causes faster drift) and vd ∝ 1/(collision frequency) (more collisions reduce net drift); thermal speed is much larger and independent of E.
Q9. A copper wire at 0°C has resistance R₀ = 10 Ω. If its resistivity increases by 4% when heated to 50°C, what is its new resistance at 50°C?
Answer: B — Since R = ρl/A and l and A remain constant, R is proportional to ρ. If ρ increases by 4%, then R increases by 4%: R = 10 × 1.04 = 10.4 Ω.
Q10. A battery maintains a steady electric field inside a conductor by continuously replenishing charge at its ends. Why is this necessary for steady current?
Answer: B — As shown in the study material, without charge replenishment (as in the initial dielectric-disc example), electrons neutralise the charges and the field vanishes, stopping the current; a battery continuously replenishes to maintain steady field and steady current.
Define electric current in terms of charge flow
Current I = ΔQ/Δt is the net charge flowing across a cross-section per unit time, in the limit of infinitesimal time interval.
What is the SI unit of current and how is it defined?
The SI unit is ampere (A), defined through the magnetic effects of currents (studied in the next chapter).
Why is there no net current in a conductor at thermal equilibrium with no applied field?
Electrons undergo random thermal motion and collide with ions; equal numbers move in all directions, so there is no preferential direction and no net charge flow.
State Ohm's law and define resistance
Ohm's law: V = IR, where V is potential difference, I is current, and R is resistance (the constant of proportionality between V and I).
How does the resistance of a conductor depend on its length and cross-sectional area?
Resistance is proportional to length (R ∝ l) and inversely proportional to cross-sectional area (R ∝ 1/A), combined as R = ρl/A.
What is drift velocity and what is its typical magnitude?
Drift velocity is the average velocity of free electrons in the direction of applied electric field; it is very small (order of mm/s) compared to random thermal speeds.
Define resistivity and state its unit
Resistivity ρ is the intrinsic resistance of a material per unit length and unit cross-sectional area; its SI unit is ohm-metre (Ω·m).
How does resistivity vary with temperature for metals and why?
Resistivity of metals increases with temperature because increased thermal motion raises collision frequency between electrons and ions, reducing drift velocity.
Distinguish between conventional current direction and electron flow direction
Conventional current is defined as the flow of positive charge and thus flows from positive to negative terminal, opposite to the actual motion of negatively charged electrons.
What role does a battery or cell play in maintaining steady current?
A battery maintains a steady electric field inside the conductor by continuously replenishing charge at the ends, ensuring a continuous current rather than transient current.
Define electric current and state its SI unit. Explain why no net current flows in a conductor at thermal equilibrium with no applied electric field. [2 marks]
Define I = ΔQ/Δt and state unit is ampere. For no net current: explain that random thermal motion gives equal electron flow in all directions, so net charge crossing any section is zero.
A uniform conductor of length l and cross-sectional area A is made of material with resistivity ρ. Derive the relation R = ρl/A by considering two identical slabs placed in series. Hence, explain how resistance depends on the geometry of the conductor. [5 marks]
Combine two identical slabs (length l each, area A) in series: total length 2l, same V across each slab, same I through both. Use Ohm's law V = IR for each slab and combination to show Rc = 2R, proving R ∝ l. Then consider parallel arrangement to show R ∝ 1/A, deriving R = ρl/A.
Explain the mechanism of current flow in a solid conductor. Describe the roles of thermal motion of electrons, collision frequency, applied electric field, and drift velocity in establishing steady current. Why is the resistivity of metals temperature-dependent? Show how a battery or cell maintains steady current in a circuit. [6 marks]
Explain: (1) Without field, random thermal motion → no net current. (2) Applied E-field gives preferential drift direction superimposed on thermal motion. (3) Drift velocity vd ∝ E and vd ∝ 1/(collision frequency). (4) Temperature ↑ → collision frequency ↑ → ρ ↑ (explain microscopically). (5) Battery maintains steady E-field inside conductor by continuously supplying charge, preventing neutralisation and keeping current steady (reference the dielectric-disc example from study material).
Practice with interactive flashcards, mind maps, upload your own chapters and get AI study kits instantly
Try StudyOS Free →