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Electricity and Magnetism
Lenz's Law
Lenz's law states that the induced electromotive force and resulting current always oppose the change in magnetic flux that produces them. It is a direct manifestation of the principle of conservation of energy in electromagnetic induction.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of the direction of induced current, application to loops entering or leaving magnetic fields, and conservation of energy implications.
PGT extension
Quantitative vector formulation involving the negative sign in Faraday's law, flux linkage calculations, motional electromotive force orientation, and connection with self and mutual induction.
Prerequisites
Concept of magnetic flux and uniform magnetic fields
Right-hand grip rule and Fleming's rules
Faraday's law of electromagnetic induction
Statement and Physical Basis of Lenz's Law
The polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it.
Mathematically represented by the negative sign in Faraday's law of induction.
It ensures that mechanical work done in moving a conductor through a magnetic field is converted into electrical energy, satisfying conservation of energy.
Conservation of Energy and Lenz's Law
If the induced current aided the flux change, it would create perpetual motion by amplifying the external cause.
Any attempt to increase flux through a coil results in an induced magnetic field pointing in the opposite direction.
Work must be continuously supplied against the magnetic retarding force to maintain current in an induction system.
Direction Determination using Right-Hand Rule
When a north pole of a magnet approaches a coil, the coil face develops a north polarity to repel the incoming magnet.
When the north pole recedes, the coil face develops a south polarity to attract the magnet and resist its withdrawal.
The direction of the induced current forms a loop that creates an opposing magnetic dipole moment.
Relation to Eddy Currents and Magnetic Damping
Lenz's law explains the creation of circulating eddy currents in bulk conductors moving in magnetic fields.
These eddy currents exert a retarding force on the moving conductor, forming the principle of electromagnetic damping.
Utilized in dead-beat galvanometers, induction furnaces, and electromagnetic braking systems.
Derivations
Conservation of Energy Verification for a Moving Rod
Conditions: A conducting rod of length l moving with velocity v in a uniform magnetic field B perpendicular to the plane of motion, connected to a resistance R.
The motional emf induced across the rod is given by e = B l v.
The magnitude of the induced current in the circuit is $i = (B l v) / R$.
The magnetic force acting on the current-carrying rod opposing its motion is $F = i l B = (B^2 l^2 v) / R$.
The mechanical power required to move the rod at constant velocity v is $P_mech = F v = (B^2 l^2 v^2) / R$.
The electrical power dissipated as heat in the resistor is $P_elec = i^2 R = ((B l v) / R)^2 R = (B^2 l^2 v^2) / R$.
Since $P_mech = P_elec$, energy is conserved, proving the validity of the negative sign mandated by Lenz's law.
Result: Mechanical power equals electrical power dissipation, confirming energy conservation through Lenz's opposing force.
Sign Convention in Faraday-Lenz Law
Conditions: Single turn flat coil of area A in a time-varying magnetic field B(t).
Magnetic flux through the coil is ph$i = B * A * \cos(\theta)$.
Faraday's experiments establish that the magnitude of induced emf is proportional to the rate of change of flux: |e| = d(phi)/dt.
To incorporate the directional opposition dictated by Lenz's law, a negative sign is introduced.
The complete equation is written as $e = - d(\phi)/dt$.
Result: $e = - d(\phi)/dt$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Faraday-Lenz Law for Single Turn
$$e = - \frac{d\phi}{dt}$$
e = induced emf, \phi = magnetic flux, t = time Volt (V)
General time-varying magnetic flux
Faraday-Lenz Law for N-Turn Coil
$$e = - N \frac{d\phi}{dt}$$
N = number of turns, e = induced emf, \phi = magnetic flux Volt (V)
Tightly wound coil where each turn experiences the same flux
Motional EMF Magnitude
$$e = B l v$$
B = magnetic field, l = length of conductor, v = velocity Volt (V)
Uniform field, mutually perpendicular vectors B, l, and v
Induced Current Magnitude
$$i = \frac{B l v}{R}$$
i = current, R = circuit resistance Ampere (A)
Closed circuit containing motional emf source of resistance R
Magnetic Force on Induced Current
$$F = \frac{B^2 l^2 v}{R}$$
F = opposing magnetic force Newton (N)
Conductor moving perpendicular to magnetic field
Self-Induction EMF
$$e = - L \frac{di}{dt}$$
$L = self-inductance$, di/dt = rate of change of current Volt (V)
Constant inductance L
Mutual Induction EMF
$$e_2 = - M \frac{di_1}{dt}$$
$e_2 = secondary emf$, M = mutual inductance, d$i_1/dt = primary current rate$ Volt (V)
magnetically coupled coils
Magnetic Flux Linkage
$$\phi = \vec{B} \cdot \vec{A} = B A \cos(\theta)$$
A = vector area, \theta = angle between B and normal to area Weber (Wb)
Uniform magnetic field over flat surface
Graphs and experiments
Magnet and Coil Relative Motion Experiment
Draw: A bar magnet with North pole facing a closed circular coil, showing deflection in a center-zero galvanometer when moved towards and away from the coil.
Interpret: Approaching North pole induces counter-clockwise current (looking from magnet side) creating North polarity to oppose entry; receding North pole induces clockwise current creating South polarity to attract and oppose exit.
Draw: An alternating current solenoid with an aluminum ring placed over its iron core.
Interpret: AC current in the solenoid induces opposing currents in the ring via Lenz's law, producing a repulsive magnetic force that causes the ring to jump into the air.
Exam use: Demonstrates dynamic application of Lenz's law under alternating magnetic fields.
Magnetic Damping of a Pendulum
Draw: A solid copper plate oscillating between the poles of an electromagnet compared to a slotted copper plate.
Interpret: Solid plate experiences heavy eddy current damping due to Lenz's law, coming to a quick halt. Slotted plate breaks eddy current paths, reducing damping.
Exam use: Examinations ask about the difference in damping behavior between solid and slotted metallic sheets.
Solved numericals
TGT core
A circular loop of radius 0.1 m is placed in a uniform magnetic field of 0.5 T normal to the plane of the loop. If the magnetic field is reduced to zero in 0.02 s, find the magnitude of the induced emf.
A rectangular wire loop of sides 0.2 m and 0.1 m has a resistance of 0.5 ohm and is pulled out of a 1.5 T magnetic field normal to the loop at a constant velocity of 2 m/s. Find the magnitude of the induced current and the mechanical force required to pull it.
Motional emf for the side cutting the field (length $l = 0.1 m): e = B * l * v = 1.5 * 0.1 * 2 = 0.3 V$.
Induced current: $i = e / R = 0.3 / 0.5 = 0.6 A$.
Opposing magnetic force: $F = i * l * B = 0.6 * 0.1 * 1.5 = 0.09 N$.
Answer: Current = 0.6 A, Force = 0.09 N
PGT extension
The current in a coil of self-inductance 5 H changes from 10 A to 2 A in 0.1 seconds. Calculate the magnitude and average direction of the induced emf relative to the main current change.
Identify parameters: L = 5 H, initial current $i_1 = 10 A$, final current $i_2 = 2 A$, dt = 0.1 s.
Find rate of change of current: di/d$t = (2 - 10) / 0.1 = -8 / 0.1 = -80 A/s$.
Physical interpretation: Since current is decreasing (di/dt < 0), the induced emf is positive relative to the current drop, attempting to maintain the current flow per Lenz's law.
Answer: 400 V opposing the decrease in current
Assumptions and limitations
Lenz's law assumes macroscopic continuity of circuits and conductors where induced charge carriers can form closed loops.
It breaks down at extremely high frequencies or quantum scales where displacement currents and quantum electrodynamic fluctuations become dominant over classical electromagnetic induction.
Common misconceptions
Common error: Lenz's law determines the magnitude of the induced electromotive force.
Correct: Lenz's law determines only the direction (polarity) of the induced emf, while Faraday's law determines its magnitude.
The magnitude is given by the rate of flux change (dphi/dt), whereas the negative sign solely dictates direction.
Common error: Lenz's law violates the principle of conservation of energy because it creates opposing forces.
Correct: Lenz's law is a direct consequence of the conservation of energy.
Mechanical work must be done against the opposing magnetic force, which directly supplies the electrical energy appearing in the circuit.
Quick revision
Lenz's law states opposition to the change in magnetic flux producing it, expressed mathematically by the negative sign in $e = -dphi/dt$.
Faraday-Lenz Law for Single Turn: $e = - \frac{d\phi}{dt}$
Faraday-Lenz Law for N-Turn Coil: $e = - N \frac{d\phi}{dt}$
Must remember
Lenz's law states that the polarity of the induced emf opposes the change in magnetic flux that produces it.
The negative sign in the Faraday-Lenz equation $e = -N \frac{d\phi}{dt}$ represents the direction mandated by Lenz's law.
Lenz's law is a direct manifestation of the principle of conservation of energy, ensuring mechanical work converts to electrical energy.
When the north pole of a magnet approaches a coil, the coil develops a north polarity to oppose the motion.
Motional EMF is given by $e = B l v$ when magnetic field, length, and velocity vectors are mutually perpendicular.
Glossary
Lenz's LawA physical law stating that induced currents flow in a direction such that their magnetic field opposes the original flux change.
Motional EMFElectromotive force induced across a conductor moving through a magnetic field, given by $e = B l v$.
Eddy CurrentsCirculating electrical currents induced in bulk conductors by changing magnetic fields, creating retarding forces via magnetic damping.
Self-InductanceThe property of a coil by which a change in its own current induces an opposing emf, denoted by $L$.
Magnetic Flux LinkageThe total magnetic flux passing through all turns of a coil, represented by $N\phi$.
Faraday's Law vs. Lenz's Law
Feature
Faraday's Law
Lenz's Law
Primary Focus
Determines magnitude of induced EMF
Determines direction (polarity) of induced EMF
Mathematical Indicator
Absolute rate of change $\frac{d\phi}{dt}$
Negative sign '$-$' in the equation
Physical Principle
Electromagnetic induction magnitude
Conservation of Energy
Self-Induction vs. Mutual Induction
Parameter
Self-Induction
Mutual Induction
Definition
EMF induced in the same coil due to change in its own current
EMF induced in a secondary coil due to current change in a primary coil
Governing Formula
$e = -L \frac{di}{dt}$
$e_2 = -M \frac{di_1}{dt}$
Coil Involvement
Single circuit loop
Two magnetically coupled circuits
Practice Quiz
Choose an option to reveal the explanation.
1. What physical principle is fundamentally upheld by Lenz's law?
Explanation: Lenz's law ensures that mechanical work done against the magnetic retarding force is converted into electrical energy, satisfying the conservation of energy.
2. The negative sign in the Faraday-Lenz equation $e = -\frac{d\phi}{dt}$ signifies:
Explanation: The negative sign is a mathematical representation of Lenz's law, showing that the induced emf opposes the change in magnetic flux.
3. If the north pole of a bar magnet is pushed towards a closed circular conducting loop, the face of the loop closest to the magnet will acquire:
Explanation: To oppose the approaching north pole, the loop develops a north polarity on the near face to repel the magnet.
4. What is the magnitude of the motional emf induced in a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$?
Explanation: The motional emf magnitude is given by the standard formula $e = B l v$ when $B$, $l$, and $v$ are mutually perpendicular.
5. Lenz's law determines which of the following properties of an induced emf?
Explanation: Lenz's law specifies only the direction or polarity of the induced emf, while Faraday's law determines its magnitude.
6. Eddy currents circulating in bulk conductors moving through magnetic fields create:
Explanation: Eddy currents generate a magnetic field that exerts a retarding force on the moving conductor, forming the principle of electromagnetic damping.
7. What is the formula for the opposing magnetic force experienced by a circuit of resistance $R$ and length $l$ moving at velocity $v$ in a field $B$?
Explanation: Substituting $i = \frac{B l v}{R}$ into the magnetic force formula $F = i l B$ yields $F = \frac{B^2 l^2 v}{R}$.
8. When the north pole of a magnet is withdrawn from a coil, the coil face develops:
Explanation: When a north pole recedes, the coil develops a south polarity to attract the magnet and resist its withdrawal, in accordance with Lenz's law.
9. What is the self-induction emf formula for a constant inductance $L$?
Explanation: Self-induction emf is given by $e = -L \frac{di}{dt}$, reflecting opposition to the change in current.
10. Which of the following devices does NOT utilize eddy currents or electromagnetic damping?
Explanation: Standard domestic resistors are designed to provide ohmic resistance and do not rely on induced eddy currents or magnetic damping.
11. What is the mutual induction emf $e_2$ in a secondary coil given a primary current rate of $\frac{di_1}{dt}$ and mutual inductance $M$?
Explanation: Mutual induction emf is expressed as $e_2 = - M \frac{di_1}{dt}$, incorporating the opposing direction mandated by Lenz's law.
12. Magnetic flux linkage $\phi$ for a uniform magnetic field vector $\vec{B}$ and vector area $\vec{A}$ at angle $\theta$ is calculated as:
Explanation: Magnetic flux linkage is defined as the dot product $\phi = \vec{B} \cdot \vec{A} = B A \cos(\theta)$.
13. If an induced current aided the flux change that produced it, what physical anomaly would occur?
Explanation: If the induced current reinforced the flux change, it would create an exponential feedback loop causing perpetual motion, violating energy conservation.
14. What is the unit of magnetic flux linkage?
Explanation: Magnetic flux and flux linkage are measured in Weber (Wb).
15. For an $N$-turn tightly wound coil where each turn experiences the same flux, the Faraday-Lenz law is expressed as:
Explanation: For $N$ identical turns sharing the same flux, the total induced emf is multiplied by $N$, giving $e = -N \frac{d\phi}{dt}$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A closed loop is moved into a magnetic field. According to Lenz's law, the direction of the induced current is such that it:
Explanation: Lenz's law dictates that the induced current always opposes the change in magnetic flux that is responsible for its generation.
2. What is the primary reason why Lenz's law must feature a negative sign in the electromagnetic induction equation $e = -\frac{d\phi}{dt}$?
Explanation: The negative sign directly reflects Lenz's law, ensuring energy is not created spontaneously, thereby upholding the conservation of energy.
3. A straight conductor of length $0.5\text{ m}$ moves at a velocity of $4\text{ m/s}$ perpendicular to a uniform magnetic field of $2\text{ T}$. What is the induced motional emf?
Explanation: Using $e = B l v$, we get $e = 2 \times 0.5 \times 4 = 4\text{ V}$.
4. Which phenomenon is directly responsible for the rapid deceleration of metal pendulums swinging through a magnetic field?
Explanation: As the metal passes through the magnetic field, eddy currents are induced which create a retarding magnetic force (electromagnetic damping).
5. A coil of $100$ turns experiences a magnetic flux change from $2\text{ Wb}$ to $0\text{ Wb}$ in $0.1\text{ s}$. What is the magnitude of the induced emf?
6. If the current in a coil with self-inductance $5\text{ H}$ changes at the rate of $2\text{ A/s}$, what is the induced self-induction emf?
Explanation: Using $e = -L \frac{di}{dt}$, we find $e = -5 \times 2 = -10\text{ V}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Lenz's law of electromagnetic induction is a consequence of the law of conservation of:
Explanation: Mechanical work required to overcome the opposing magnetic force transforms into electrical energy, satisfying conservation of energy.
2. A metal ring is placed above a solenoid carrying an alternating current. The ring floats due to:
Explanation: The alternating flux from the solenoid induces currents in the ring that oppose the primary flux, creating a repulsive magnetic force.
3. The direction of induced current in a loop when magnetic flux increases into the page is such that its magnetic field points:
Explanation: To oppose the increase in inward flux, the induced current creates a magnetic field pointing out of the page.
4. Calculate the magnitude of the opposing magnetic force on a circuit where $B = 0.5\text{ T}$, $l = 0.4\text{ m}$, $v = 2\text{ m/s}$, and $R = 0.8\;\Omega$.
5. In electromagnetic damping systems, why are slots often cut into copper sheets moving through magnetic fields?
Explanation: Slits restrict the paths of eddy currents, optimizing the retarding magnetic force for precise damping without excessive heating.
6. A uniform magnetic field $B$ is perpendicular to a circular loop of radius $r$. If the field decreases at rate $\frac{dB}{dt}$, the induced emf is:
Explanation: Flux is $\phi = B \pi r^2$. Since $B$ decreases, $-\frac{d\phi}{dt} = -(\pi r^2)\frac{dB}{dt}$ in magnitude terms, but if the field is decreasing, the negative sign of the rate combines with the formula to yield a positive magnitude or positive orientation depending on context; standard magnitude is $\pi r^2 \frac{dB}{dt}$.
7. Mutual inductance between two coils depends upon:
Explanation: Mutual inductance $M$ is governed by coil geometry, proximity, number of turns, and core material properties.
8. Which rule is commonly used to determine the polarity of a coil face under Lenz's law when a magnet approaches?
Explanation: The Right-Hand Rule establishes that the direction of induced current forms a loop creating an opposing magnetic dipole moment.
9. If a bar magnet falls vertically through a hollow vertical copper tube, its acceleration is:
Explanation: Eddy currents induced in the copper tube exert an upward magnetic retarding force on the falling magnet, reducing its net acceleration below $g$.
10. What happens to the induced current in a stationary closed loop if a constant, unchanging magnetic field passes through it?
Explanation: An induced emf and current require a *change* in magnetic flux ($\frac{d\phi}{dt} \neq 0$). A constant flux produces zero induced current.