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Electricity and Magnetism
Electric Current and Current Density
Electric current is defined as the rate of flow of net electric charge through a given cross-section of a conductor. Microscopic analysis relates macroscopic current to the drift velocity of free charge carriers within the conductor lattice.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of electric current, Ohm's law, resistance, resistivity, grouping of resistors, and simple DC circuits.
PGT extension
Drift velocity derivation, microscopic form of Ohm's law (J = σE), temperature dependence of resistivity, and complex network analysis using Kirchhoff's laws.
Prerequisites
Conservation of charge
Electrostatic potential and electric field inside conductors
Basic vector calculus and scalar product
Definition and Nature of Electric Current
Electric current is a scalar quantity defined as the rate of flow of charge: $I = dQ/dt$.
The SI unit of electric current is the Ampere (A), which is a fundamental SI unit equal to one Coulomb per second.
Even though it has magnitude and direction, current does not obey the parallelogram law of vector addition, hence it is classified as a scalar.
Microscopic Model and Drift Velocity
Free electrons in a conductor move randomly with thermal velocities of the order of 10&sup5; m/s.
When an electric field E is applied, electrons acquire a small net directed velocity opposite to the field, known as drift velocity (v&subscript;d).
Drift velocity is typically of the order of 10&sup-&sup4; m/s, which is extremely small compared to thermal velocity or the speed of propagation of the electric field.
Current Density and Vector Form of Ohm's Law
Current density (J) is a vector quantity defined as current per unit cross-sectional area normal to the direction of flow: $J = I/A = ne v_{script}d$.
The SI unit of current density is Ampere per square meter (A/m²).
The microscopic form of Ohm's law relates current density to the electric field via conductivity (σ): $J = \sigmaE$.
Mobility of Charge Carriers
Mobility (μ) is defined as the magnitude of drift velocity per unit electric field: $\mu = v_{script}d / E$.
The SI unit of mobility is m²/(V·s).
Conductivity can be expressed in terms of carrier concentration (n), charge of electron (e), and mobility (μ) as $\sigma = ne\mu$.
Derivations
Relation between Electric Current and Drift Velocity
Conditions: Uniform conductor of cross-sectional area A, uniform free electron density n, and uniform drift velocity v&subscript;d.
Consider a cylinder of length v&subscript;d dt inside a conductor through which electrons drift in time dt.
The volume of this cylinder is A v&subscript;d dt, and the number of free electrons contained in it is $N = n A v_{script}d dt$.
The total charge contained in this volume is d$Q = N e = n e A v_{script}d dt$.
The current is given by $I = dQ / dt = n e A v_{script}d$.
Result: $I = neAv_d$
Derivation of Microscopic Ohm's Law from Drude Model
Conditions: Conductor under steady electric field E, relaxation time τ.
Result: J = \sigma E
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Electric Current Definition
$$I = \frac{dQ}{dt}$$
I = Current, Q = Charge, t = Time A (Ampere)
General time-varying or steady current
Current in terms of Drift Velocity
$$I = neAv_d$$
n = electron density, e = electronic charge, A = area, $v_d = drift velocity$ A
Uniform conductor and steady state
Current Density
$$J = \frac{I}{A} = ne v_d$$
J = Current density, I = Current, A = Area A/m^2
Normal cross-section
Microscopic Ohm's Law
$$J = \sigma E$$
J = Current density, \sigma = Conductivity, E = Electric field A/m^2
Isotropic and linear media
Electrical Resistivity and Resistance
$$R = \rho \frac{l}{A}$$
R = Resistance, \rho = Resistivity, l = length, $A = cross-sectional area$ \Omega
Uniform cross-section
Temperature Dependence of Resistivity
$$\rho_T = \rho_0 [1 + \alpha(T - T_0)]$$
\rh$o_T = resistivity at T$, \rh$o_0 = resistivity at T_0$, \alpha = temperature coefficient \Omega\cdot\text{m}
Moderate temperature ranges
Mobility of Charge Carriers
$$\mu = \frac{v_d}{E}$$
\mu = mobility, $v_d = drift velocity$, E = electric field m^2/(V\cdot\text{s})
Uniform electric field
Drift Velocity Expression
$$v_d = \frac{eE\tau}{m}$$
e = charge, E = field, \tau = relaxation time, m = mass m/s
Drude model assumptions
Graphs and experiments
V-I Characteristic Graph for Metallic Conductor
Draw: A straight line passing through the origin on a V (voltage) versus I (current) plot at constant temperature.
Interpret: The slope of the V-I graph (dV/di) represents the electrical resistance R of the conductor.
Exam use: Frequently asked in TGT/PGT to identify ohmic devices and calculate resistance from graphical slopes.
Resistivity vs Temperature for Metals
Draw: A curve that starts near zero at absolute zero and increases almost linearly at higher temperatures, curving slightly at lower temperatures.
Interpret: Shows that resistance increases with temperature due to increased frequency of atomic collisions reducing relaxation time.
Exam use: Distinguishing between metallic conductors and semiconductors (which show negative temperature coefficient).
Meter Bridge Experiment (Verification of Laws of Resistance)
Draw: Schematic setup showing a 1-meter resistance wire, resistance box, unknown resistor, galvanometer, and jockey.
Interpret: Based on Wheatstone bridge principle at balance point: R/$S = l_1 / (100 - l_1)$.
Exam use: Testing experimental determination of unknown resistance and specific resistance.
Solved numericals
TGT core
A current of 2 A flows through a uniform wire of cross-sectional area 1 mm^2. If the free electron density in the wire is 10^28 m^-3, calculate the drift velocity of the electrons.
Identify given values: I = 2 A, $A = 1 mm^2 = 10^-6 m^2$, $n = 10^28 m^-3$, $e = 1.6 \times 10^-19 C$.
Use the formula for current: $I = neAv_d$.
Rearrange to solve for drift velocity: $v_d = I / (neA)$.
An aluminum wire of diameter 2 mm and length 3 m is connected across a potential difference of 2 V. If the resistivity of aluminum is 2.8 \times 10^-8 \Omega\cdot\text{m}, find the current density inside the wire.
The resistance of a metallic wire at 20 \deg\text{C} is 20 \Omega. If the temperature coefficient of resistance is 0.004 /\deg\text{C}, at what temperature will the resistance become 40 \Omega?
Use formula $R_T = R_0 [1 + \alpha(T - T_0)]$.
Given $R_0 = 20 \Omega at T_0 = 20 \deg\text{C}$, $R_T = 40 \Omega$, $\alpha = 0.004 /\deg\text{C}$.
Solve for T: T - $20 = 1 / 0.004 = 250 \implies T = 270 \deg\text{C}$.
Answer: 270 \deg\text{C}
Assumptions and limitations
Ohm's law assumes that physical conditions such as temperature and mechanical strain remain constant during current flow.
The Drude model assumes classical mechanics apply to electrons, treating them as an ideal gas colliding with fixed heavy ions, which fails at very low temperatures where quantum effects dominate.
Common misconceptions
Common error: Electric current flows at the speed of light inside a conductor.
Correct: The electric field propagates at nearly the speed of light, but the actual drift velocity of electrons is extremely small (order of 10^-4 m/s).
Electrons undergo continuous random collisions and drift slowly under the electric field.
Common error: Current is a vector quantity because it has both magnitude and direction.
Correct: Current is a scalar quantity because it does not follow the vector addition laws (parallelogram law).
The sum of currents at a junction depends only on algebraic addition (Kirchhoff's Current Law).
Common error: Resistance depends on the applied potential difference and current.
Correct: Resistance is an intrinsic property of the conductor geometry and material ($R = \rho l / A)$, independent of V and I for ohmic conductors.
Ratio V/I remains constant for ohmic materials irrespective of individual changes in V or I.
Quick revision
Current is the scalar rate of flow of charge: $I = dQ/dt$.
Microscopic current relation: $I = neAv_d$.
Vector form of Ohm's law: J = \sigma E.
Conductivity and resistivity are reciprocal: $\sigma = 1 / \rho$.
Drift velocity is directly proportional to the applied electric field: $v_d = \mu E$.
Resistance varies with temperature according to $R_T = R_0 [1 + \alpha \Delta T]$.
Must remember
Electric current ($I$) is a scalar quantity defined as the rate of flow of charge: $I = \frac{dQ}{dt}$.
Drift velocity ($v_d$) of electrons in a conductor is typically of the order of $10^{-4}\text{ m/s}$, whereas thermal velocity is around $10^5\text{ m/s}$.
Current density ($J$) is a vector quantity given by $J = \frac{I}{A} = ne v_d$.
Microscopic form of Ohm's law connects current density to electric field and conductivity: $J = \sigma E$.
Mobility of charge carriers is defined as $\mu = \frac{v_d}{E}$ with SI unit $\text{m}^2/(\text{V}\cdot\text{s})$.
Glossary
Electric CurrentThe rate of flow of electric charge across a cross-sectional area, measured in Amperes.
Drift VelocityThe average uniform velocity acquired by free electrons inside a conductor under the influence of an external electric field.
Current DensityElectric current passing per unit cross-sectional area normal to the direction of current flow.
MobilityThe magnitude of drift velocity per unit electric field for charge carriers.
Microscopic Ohm's LawThe vector relation $J = \sigma E$ valid at any point inside a conducting medium.
Thermal Velocity vs. Drift Velocity
Parameter
Thermal Velocity
Drift Velocity
Order of magnitude
$10^5\text{ m/s}$
$10^{-4}\text{ m/s}$
Direction
Random in all directions
Directed opposite to electric field
Cause
Thermal energy of electrons
Applied external electric field
Electric Current vs. Current Density
Property
Electric Current ($I$)
Current Density ($J$)
Nature
Scalar quantity
Vector quantity
SI Unit
Ampere (A)
$\text{A/m}^2$
Definition
Rate of flow of charge
Current per unit normal area
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following physical quantities is a scalar despite having magnitude and direction?
Explanation: Electric current has magnitude and direction but does not obey the parallelogram law of vector addition, making it a scalar quantity.
2. What is the SI unit of electric current?
Explanation: The SI unit of electric current is the Ampere (A), which is a fundamental SI unit.
3. What is the typical order of magnitude of thermal velocities of free electrons in a conductor at room temperature?
Explanation: Free electrons move randomly due to thermal agitation with velocities of the order of $10^5\text{ m/s}$.
4. The drift velocity $v_d$ of free electrons in a metallic conductor subjected to an electric field $E$ is of the order of:
Explanation: Under an electric field, electrons drift very slowly with a speed typically around $10^{-4}\text{ m/s}$.
5. Current density $J$ is related to drift velocity $v_d$ and electron concentration $n$ by the expression:
Explanation: Current density is given by $J = \frac{I}{A} = ne v_d$.
6. Which expression represents the microscopic form of Ohm's law?
Explanation: The microscopic form of Ohm's law relates current density $J$ to the electric field $E$ and conductivity $\sigma$ via $J = \sigma E$.
7. The SI unit of current density is:
Explanation: Current density is current per unit cross-sectional area normal to flow, so its unit is $\text{A/m}^2$.
8. Mobility of charge carriers $\mu$ is defined as:
Explanation: Mobility is the magnitude of drift velocity per unit electric field: $\mu = \frac{v_d}{E}$.
9. What is the SI unit of mobility?
Explanation: Since $\mu = v_d / E$, its unit is $(\text{m/s}) / (\text{V/m}) = \text{m}^2/(\text{V}\cdot\text{s})$.
10. Electrical conductivity $\sigma$ can be expressed in terms of carrier concentration $n$, electronic charge $e$, and mobility $\mu$ as:
Explanation: Using $J = \sigma E$ and $J = ne v_d = ne (\mu E)$, we get $\sigma = ne\mu$.
11. According to the Drude model, drift velocity is proportional to:
Explanation: Drift velocity is given by $v_d = \frac{eE\tau}{m}$, showing it is directly proportional to the relaxation time $\tau$.
12. Why does electric current not obey vector addition laws?
Explanation: Current follows Kirchhoff's Current Law, which is a simple algebraic sum independent of the angle between wires.
13. What happens to the drift velocity of electrons when the magnitude of the applied electric field is doubled?
Explanation: Since $v_d = \frac{eE\tau}{m}$, drift velocity is directly proportional to the electric field $E$.
14. The resistance $R$ of a uniform conductor of length $l$ and cross-sectional area $A$ is given by:
Explanation: Resistance is directly proportional to length and inversely proportional to cross-sectional area: $R = \rho \frac{l}{A}$.
15. How does the resistivity of a typical metallic conductor vary with an increase in temperature?
Explanation: For metals, resistivity increases with temperature according to $\rho_T = \rho_0[1 + \alpha(T - T_0)]$ with positive temperature coefficient $\alpha$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A current $I$ flows through a uniform wire of radius $r$. If the drift velocity of electrons is $v_d$, what is the drift velocity when a wire of radius $2r$ made of the same material carries the same current?
Explanation: Current $I = neA v_d = ne(\pi r^2) v_d$. For the same current and material ($n, e$ constant), $r^2 v_d$ is constant. If radius becomes $2r$, area becomes $4$ times, so drift velocity becomes $v_d/4$.
2. The electric field $E$, current density $J$, and conductivity $\sigma$ of a conductor are related by:
Explanation: The microscopic form of Ohm's law is $J = \sigma E$.
3. If the number density of free electrons in a copper wire is $8.5 \times 10^{28}\text{ m}^{-3}$ and the area of cross-section is $10^{-6}\text{ m}^2$ carrying a current of $1.7\text{ A}$, the drift velocity of electrons is:
6. A potential difference $V$ is applied across a conductor of length $l$ and uniform cross-section $A$. The drift velocity of electrons is proportional to:
Explanation: Since $v_d = \frac{eE\tau}{m}$ and $E = \frac{V}{l}$, drift velocity is proportional to $V/l$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If a charge of $60\text{ C}$ passes through a wire in $2\text{ minutes}$, the steady electric current flowing through the wire is:
2. Current density vector $J$ and electric field vector $E$ are related by $\sigma$. For an isotropic conductor, $J$ and $E$ are:
Explanation: Since $J = \sigma E$ and $\sigma$ is a scalar for isotropic media, $J$ and $E$ are in the same direction (parallel).
3. The drift speed of electrons in a copper wire of length $2\text{ m}$ under a potential difference of $10\text{ V}$ is $v_d$. If the potential difference is increased to $20\text{ V}$ keeping the length constant, the new drift speed will be:
Explanation: Drift velocity is directly proportional to the applied electric field (or potential difference for fixed length), so doubling $V$ doubles $v_d$.
4. What is the primary reason for the very high speed of electrical signals in a circuit compared to the drift speed of electrons?
Explanation: The electric field sets up almost instantaneously across the entire circuit at the speed of light, causing all free electrons to drift simultaneously.
5. If the resistivity of a material is $\rho$ and number density of electrons is $n$, the relaxation time $\tau$ can be expressed as:
Explanation: Conductivity $\sigma = \frac{ne^2\tau}{m}$, and since $\rho = \frac{1}{\sigma}$, we have $\rho = \frac{m}{ne^2\tau}$.
6. A wire of resistance $R$ is stretched to double its original length while keeping its volume constant. Its new resistance will be:
Explanation: When a wire is stretched to $n$ times its length at constant volume, its resistance becomes $n^2 R$. Here $n = 2$, so new resistance is $4R$.
7. The temperature coefficient of resistance for a semiconductor is generally:
Explanation: Semiconductors have a negative temperature coefficient of resistance because carrier concentration increases significantly with temperature, lowering resistivity.
8. Which of the following statements is correct regarding current density?
Explanation: Current density $J = \frac{I}{A}$ explicitly depends on the cross-sectional area normal to the direction of current flow.
9. The drift velocity of electrons depends on:
Explanation: From $v_d = \frac{eE\tau}{m}$, drift velocity depends on both the electric field $E$ and the relaxation time $\tau$.
10. In a cylindrical conductor of radius $r$ and length $l$, the current density is uniformly distributed across its cross-section. If total current is $I$, the magnitude of current density $J$ is:
Explanation: Current density $J = \frac{I}{A} = \frac{I}{\pi r^2}$ since the cross-sectional area is $\pi r^2$.