Introduction

In the previous discussions on electromagnetic induction, we have qualitatively understood that Lenz’s law is a direct consequence of the law of conservation of energy. The opposing direction of the induced current ensures that energy is neither created nor destroyed but transformed from one form to another.

In this section, we will examine this idea quantitatively through a concrete example involving a movable conductor in a magnetic field. This analysis helps establish that the mechanical energy expended to move a conductor through a magnetic field is converted into electrical energy (induced emf) and subsequently dissipated as thermal energy (Joule heat) in the circuit.

The Experimental Setup

PQRS is a rectangular conducting loop that is placed in a spatially homogeneous field of magnetic induction B, and perpendicular to the plane of the page (see Figure 1).

PQ is a transversely movable conductor of scope L that can freely move, without resistance, along the parallel conducting tracks QR and SP.
The stationary segment of conductor RS, closes the electrical path, and thus forms a geometrically closed rectangular current loop.

Suppose the conditions as follows:

The moving arm PQ has an electrical resistance r,
The rest of the loops (QR, RS, and SP) are assumed to have an insignificant resistance,
The arm PQ is moved off to the right at a constant velocity v.

The magnetic induction is at right angles to the plane of the loop and the circuit connectivity is maintained during the motion of PQ. This motion of the rod through this magnetic field creates a motional electromotive force that conveys a current of induced electricity all over the loop.

Induced EMF and Current in the Loop

As the arm PQ travels with velocity v, perpendicular to the magnetic field B, the carriers of charge in PQ undergo a Lorentz force due to this motion in the magnetic field. This force causes a spatial separation in charges whereby one end of the rod ends up possessing a net positive charge and the other end ends up possessing a net negative charge thus creating an induced electromotive force (emf).

As defined in the law of Faraday, induced emf is given as:

ε = BLv

Ohm's law, emf provides a current through the circuit governed by the law:

I = ε/r = BLv/r

The induced current flows in the direction which causes the resulting magnetic field to be opposite to the change of magnetic flux, which caused the current, the basic fact of the law of Lenz.

Magnetic Force on the Moving Conductor

Taking into account that the rod PQ currently contains a current of magnitude I in a magnetic field B, the rod PQ undergoes magnetic Lorentz force. The right-hand rule can be used to determine the direction of this force: as the motion of PQ tends, this force acts opposite to the motion, and, therefore, in the opposite direction to the external force which causes this motion.

The magnitude of the magnetic force is:

F = ILB

Substituting the expression for I:

F = B²L²v/r

This force opposes the movement of the conductor and therefore, to ensure that the speed remains constant at a speed v, mechanical work must be provided.

Mechanical Power and Electrical Dissipation

The external agent (say, a hand or motor) that moves the arm PQ must apply a constant external force equal and opposite to F to keep the velocity constant. The power required to do this is:

P = Fv = B²L²v²/r

This mechanical power is supplied continuously by the external agent. Now, due to the current I flowing in the circuit, energy is dissipated as Joule heat in the resistor PQ.

The rate of Joule heat dissipation is:

P_j = I²r = (BLv/r)²r = B²L²v²/r

We observe that:

P = P_j

Thus, the mechanical power input equals the electrical power dissipated. This confirms that no energy is lost: mechanical energy is completely converted into thermal energy, consistent with the law of conservation of energy.

Energy Conservation Verification

The equality P = P_j is not merely a mathematical coincidence—it demonstrates the fundamental principle that governs all electromagnetic processes.

The agent’s mechanical work done in moving the rod is converted to electrical energy through induction.
The electrical energy in turn is dissipated as heat due to resistance in the circuit.

Hence, Lenz’s law ensures that energy conservation is obeyed. If the induced current aided the motion (instead of opposing it), the rod would accelerate spontaneously without any external input—violating conservation of energy.

Relation Between Induced Charge and Magnetic Flux

There exists a fascinating relationship between the total charge flow in the circuit and the change in magnetic flux through it.

From Faraday’s law:

|ε| = ΔΦ_B/Δt

and from Ohm’s law:

|ε| = Ir = (ΔQ/Δt) × r

Equating the two:

ΔΦ_B/Δt = (ΔQ/Δt) × r

Thus,

ΔQ = ΔΦ_B/r

This expression implies that the total charge passing through any cross-section of the circuit during a change in magnetic flux depends directly on the magnitude of flux change and inversely on the resistance.

Example: Motion of Arm PQ

Let us now apply these principles to a specific situation.

Problem setup:

The arm PQ of a rectangular conductor is moved outward from x=0 to x=2b (Fig. 2 (a)).
The uniform magnetic field exists only in the region 0<x<b, and is zero for x>b.
The resistance of the arm PQ is r, and the rod moves with constant speed v.

We are to determine:

The flux, induced emf, force, and Joule heat loss for the motion.
Their variation with distance during outward and inward motion.

Magnetic Flux and Induced EMF

The magnetic flux linked with the circuit when the rod is at position x is given by:

Φ_B = BLx (0 ≤ x ≤ b)

Beyond x=b, the flux remains constant since the rod has left the field region:

Φ_B = BLb (b ≤ x < 2b)

Hence, the induced emf is:

ε = –dΦ_B/dt

Substituting x=vt:

ε = –BLv for 0 < x < b

and

ε = 0 for b < x < 2b

This means the emf is non-zero only while PQ is within the magnetic field region.

Induced Current During Motion

The current induced in the loop is:

I = |ε|/r = BLv/r

For 0 < x < b: Current flows in one direction (say, clockwise).
For b < x < 2b: No emf is induced, so I=0.

When PQ moves back from x=2b to x=0, the induced emf reverses sign, and the current flows in the opposite direction (anticlockwise).

Magnetic Force and Power Dissipation

The magnetic force on the rod (Fig. 2 (b)) during outward motion is:

F = ILB = B²L²v/r

and it acts to the left, opposing the motion.

For the region outside the magnetic field (b < x < 2b), F=0.

The Joule heating loss during motion is:

P_j = I²r = B²L²v²/r

This remains constant as long as the rod is within the field region and zero outside it.

Graphical Interpretation (Fig. 2 (b))

The variation of various physical quantities with the rod’s position x can be represented graphically:

Quantity	0 < x < b	b < x < 2b
Flux Φ_B	Increases linearly with x	Constant
EMF ε	Constant (–BLv)	Zero
Force F	Constant (–B²L²v/r)	Zero
Power P_j	Constant (B²L²v²/r)	Zero

During inward motion (from 2b→0), all quantities reverse direction (sign), but magnitudes remain the same.

Thus, as shown in Fig. 2 (b), the flux, emf, force, and power exhibit piecewise constant or linear variations with distance.

Reverse Motion Analysis

During the return journey (from x=2b to x=0), similar expressions hold true, except that the signs reverse:

ε = +BLv, F = –B²L²v/r, P_j = B²L²v²/r

Thus, the magnitude of emf, current, force, and power remain the same as before, but the direction of current and force are reversed.

The total mechanical energy supplied during the complete round trip equals the total heat dissipated in the resistor, reaffirming energy conservation.

Discussion and Physical Interpretation

Nature of Force:
The magnetic force on the conductor acts opposite to the direction of motion, demonstrating Lenz’s law dynamically.
Energy Flow:
- The external agent provides mechanical energy.
- This is converted into electrical energy (due to emf).
- The electrical energy is dissipated as Joule heat in the resistor.
No Energy Stored:
Since the system returns to its initial configuration after motion, no net electrical or magnetic energy remains stored. The entire mechanical work is transformed into heat.
Universality:
The same principle operates in electric generators, where mechanical work is continuously converted into electrical energy, with energy losses occurring due to resistances and mechanical friction.

Energy Balance Equation

For any infinitesimal motion dx of PQ:

Work done by external agent = Energy dissipated as heat

F dx = I²r dt

Substituting dx = v dt and F = B²L²v/r:

(B²L²v/r) × v dt = (B²L²v²/r) × dt

This equality holds at every instant of motion, proving that mechanical energy is instantaneously and completely converted into thermal energy.

Applications and Significance

Electric generators: Mechanical work from turbines is converted into electrical energy following the same principle.
Electromagnetic braking: Kinetic energy of moving wheels or rotors is converted into heat through induced currents.
Measurement devices: Understanding induced emf and current flow allows precise design of galvanometers and induction-based sensors.

This quantitative study thus not only confirms Lenz’s law but also emphasizes its vital role in practical electromagnetic systems.

Conclusion

The quantitative analysis of energy consideration in electromagnetic induction provides a vivid demonstration of the law of conservation of energy. The work done in moving a conductor through a magnetic field is completely converted into heat through the induced current.

Through the example of the moving arm PQ:

The induced emf, current, force, and power relations were derived.
The equivalence of mechanical power input and Joule power output was established.
The process of energy conversion was understood graphically and mathematically.

Thus, Lenz’s law emerges as a natural and necessary outcome of the universal law of energy conservation — ensuring that energy merely changes its form but is never lost.

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