Electromagnetic Induction Basics: Faraday and Lenz Laws

The Experiments Of Faraday And Henry

For a long time, scientists knew that electric currents could produce magnetic fields. The reverse effect – whether magnetic fields could produce electric currents – was a subject of investigation. In 1831, independently and almost simultaneously, Michael Faraday in England and Joseph Henry in the USA discovered that varying magnetic fields can induce electric currents in nearby conductors. This phenomenon is called electromagnetic induction.

Let's describe some of the key experiments conducted by Faraday (and Henry) that demonstrated electromagnetic induction.

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Experiment 1: Moving Magnet and Coil

Setup: A coil of wire is connected to a galvanometer, which is a sensitive instrument to detect current. A bar magnet is used. The coil and the galvanometer form a closed circuit, but there is no battery or external power source connected.

Faraday's Experiment 1: Observing induced current with a moving magnet.

Observations:

• When the North pole of the bar magnet is pushed towards the coil, the galvanometer shows a momentary deflection, indicating the presence of an electric current in the coil. The current flows as long as the magnet is moving.
• When the North pole is pulled away from the coil, the galvanometer shows a momentary deflection again, but in the opposite direction, indicating that the induced current flows in the reverse direction.
• When the magnet is held stationary inside or near the coil, there is no deflection in the galvanometer, meaning no current is induced.
• If the South pole of the magnet is moved towards or away from the coil, similar effects are observed, but the direction of the induced current is reversed compared to the movement of the North pole.
• Moving the coil towards or away from the stationary magnet also induces a current, with the direction depending on the relative motion.
• The faster the relative motion between the magnet and the coil, the greater the deflection in the galvanometer, indicating a larger induced current.

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Experiment 2: Moving Coil in a Magnetic Field

Setup: A coil is connected to a galvanometer. It is placed between the poles of a magnet where there is a magnetic field.

Faraday's Experiment 2: Observing induced current by moving a coil in a magnetic field.

Observations:

• When the coil is moved into or out of the magnetic field, the galvanometer shows a momentary deflection.
• When the coil is moved within the uniform part of the magnetic field, or held stationary, there is no deflection.
• Moving the coil faster results in a larger deflection.

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Experiment 3: Varying Current in a Neighbouring Coil

Setup: Two coils, a primary coil (P) and a secondary coil (S), are placed near each other. The primary coil is connected to a battery and a switch (or a rheostat). The secondary coil is connected to a galvanometer. There is no direct electrical connection between the two coils.

Faraday's Experiment 3: Observing induced current due to varying current in a nearby coil.

Observations:

• When the switch in the primary circuit is closed (current starts flowing in P), the galvanometer in the secondary circuit shows a momentary deflection.
• When the current in the primary coil is steady (switch is kept closed), there is no deflection in the galvanometer.
• When the switch in the primary circuit is opened (current stops flowing in P), the galvanometer shows a momentary deflection again, but in the opposite direction to the first deflection.
• If a rheostat is used in the primary circuit to change the current gradually, the galvanometer shows a continuous deflection as long as the current is changing. The faster the current is changed, the larger the deflection.
• Moving one coil relative to the other while current is flowing in the primary also induces a current in the secondary.

From these experiments, Faraday concluded that an electric current is induced in a coil whenever there is a change in the magnetic conditions around it. It is not the mere presence of a magnetic field, but the change in the magnetic field or the magnetic flux, that causes the induction of current and EMF.

Magnetic Flux ($ \Phi_B = \int \vec{B} \cdot d\vec{A} $)

To quantify the "magnetic conditions" around a coil and make the concept of "change in magnetic conditions" more precise, Faraday introduced the concept of magnetic flux.

Magnetic flux ($\Phi_B$) through a given surface is defined as the total number of magnetic field lines passing normally through that surface. It is a scalar quantity.

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Definition of Magnetic Flux

Consider a small planar area element $d\vec{A}$ located in a magnetic field $\vec{B}$. The area vector $d\vec{A}$ is a vector perpendicular to the surface element and has a magnitude equal to the area $dA$. The magnetic flux $d\Phi_B$ through this small element is defined as the dot product of the magnetic field and the area vector:

$ d\Phi_B = \vec{B} \cdot d\vec{A} = B \, dA \cos\theta $

where $\theta$ is the angle between the magnetic field vector $\vec{B}$ and the area vector $d\vec{A}$.

Magnetic flux through a surface element $d\vec{A}$.

For a finite surface S, the total magnetic flux $\Phi_B$ is obtained by integrating the magnetic flux through all the infinitesimal area elements over the entire surface:

$ \Phi_B = \int_S \vec{B} \cdot d\vec{A} $

If the magnetic field $\vec{B}$ is uniform over a flat surface with total area $\vec{A}$, the integral simplifies to the dot product:

$ \Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta $

where $\theta$ is the angle between $\vec{B}$ and $\vec{A}$.

For a coil consisting of $N$ turns, if the same magnetic flux $\Phi_B$ passes through each turn, the total flux linked with the coil is $N\Phi_B$.

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Units of Magnetic Flux

The SI unit of magnetic flux is the weber (Wb). From the formula $\Phi_B = BA$, the unit Weber can also be expressed as Tesla-meter squared ($T \cdot m^2$).

$ 1 \text{ Wb} = 1 \text{ T} \cdot m^2 $

The magnetic field $\vec{B}$ is also sometimes called the magnetic flux density, as its unit (Tesla) is equivalent to Wb/m$^2$.

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Factors Affecting Magnetic Flux

The magnetic flux through a surface can be changed by varying any of the factors in the integral $\int_S \vec{B} \cdot d\vec{A}$:

1. Change in magnetic field strength ($B$): If the magnitude of the magnetic field changes with time or location. (e.g., moving a magnet, changing current in a nearby coil).
2. Change in area ($A$): If the area of the loop or coil changes in the magnetic field. (e.g., deforming a flexible loop in a magnetic field).
3. Change in orientation ($\theta$): If the angle between the magnetic field and the area vector of the surface changes. (e.g., rotating a coil in a magnetic field).

Faraday's experiments demonstrated that it is precisely this change in magnetic flux that induces an EMF and hence current in a circuit.

Faraday’S Law Of Induction ($ \mathcal{E} = -\frac{d\Phi_B}{dt} $)

Based on his experiments, Michael Faraday formulated the quantitative laws of electromagnetic induction. These laws establish the relationship between the change in magnetic flux and the induced EMF.

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Faraday's First Law of Induction

This law states that:

"Whenever there is a change in the magnetic flux linked with a coil, an electromotive force (EMF) is induced in the coil."

"The induced EMF lasts only as long as the change in magnetic flux continues."

This law explains the basic phenomenon observed in Faraday's experiments – the momentary deflection of the galvanometer when the magnetic flux changes. No change in flux means no induced EMF or current.

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Faraday's Second Law of Induction

This law quantifies the magnitude of the induced EMF:

"The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linked with the coil."

If $\Phi_B$ is the magnetic flux linked with a single turn of a coil at any instant, and the flux changes by $d\Phi_B$ in a time interval $dt$, the instantaneous rate of change of flux is $d\Phi_B/dt$. According to the law, the magnitude of the induced EMF $|\mathcal{E}|$ is proportional to this rate:

$ |\mathcal{E}| \propto \frac{d\Phi_B}{dt} $

In SI units, the constant of proportionality is 1. So, for a single turn:

$ |\mathcal{E}| = \frac{d\Phi_B}{dt} $

If the coil has $N$ turns, and the flux linked with each turn is the same $\Phi_B$, the total flux linkage with the coil is $N\Phi_B$. The induced EMF in the coil is proportional to the rate of change of the total flux linkage:

$ |\mathcal{E}| = \frac{d(N\Phi_B)}{dt} $

If $N$ is constant, this becomes:

$ |\mathcal{E}| = N \left|\frac{d\Phi_B}{dt}\right| $

This formula gives the magnitude of the induced EMF. The induced current ($I$) in the circuit, if it has resistance $R$, is given by $I = |\mathcal{E}| / R$.

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Faraday's Law (Combined Form with Direction - Lenz's Law)

The direction of the induced EMF (and hence the induced current) is given by Lenz's Law, which incorporates the negative sign in the mathematical formulation of Faraday's Law.

Faraday's Law, including the direction, is written as:

$ \mathcal{E} = -\frac{d\Phi_B}{dt} $ (for a single turn)

Or, for a coil with $N$ turns:

$ \mathcal{E} = -N \frac{d\Phi_B}{dt} = -\frac{d(N\Phi_B)}{dt} $

The negative sign indicates the direction of the induced EMF, as explained by Lenz's Law.

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Ways to Change Magnetic Flux

The magnetic flux linked with a circuit can change in several ways:

1. Changing the magnetic field $\vec{B}$: Moving a magnet towards or away from a coil, or changing the current in a primary coil near a secondary coil.
2. Changing the area $\vec{A}$ of the circuit: Deforming a loop in a magnetic field, or moving a conductor such that the effective area perpendicular to the field changes.
3. Changing the angle $\theta$ between $\vec{B}$ and $\vec{A}$: Rotating a coil in a magnetic field (this is the principle behind electric generators).
4. Any combination of the above.

In all these cases, a change in magnetic flux induces an EMF. The induced EMF causes current to flow if the circuit is closed.

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Answer:

Lenz’S Law And Conservation Of Energy

Faraday's Laws tell us the magnitude of the induced EMF. However, they do not specify the direction of the induced current. This crucial aspect is provided by Lenz's Law, formulated by Heinrich Lenz in 1834.

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Statement of Lenz's Law

Lenz's Law states that:

"The direction of the induced EMF or induced current in a circuit is such that it opposes the very cause that produces it."

The "cause" is the change in magnetic flux. So, the induced current will flow in a direction that creates a magnetic field opposing the change in flux.

Let's revisit Faraday's Experiment 1 (Moving Magnet and Coil) using Lenz's Law:

• North pole of a magnet is pushed towards the coil: The magnetic flux linked with the coil increases in the direction of the magnetic field lines of the North pole (outwards from N). The induced current will flow in a direction such that it produces a magnetic field opposing this increase. To oppose the increase in outward flux, the induced current must create an inward magnetic field. Using the Right-Hand Rule for a coil, the current must flow in a direction that makes the coil behave like a North pole facing the approaching magnet (repulsion). This induced North pole pushes against the approaching North pole of the magnet, opposing its motion.
• North pole of a magnet is pulled away from the coil: The magnetic flux linked with the coil decreases in the direction of the magnetic field lines of the North pole (outwards). The induced current will flow in a direction such that it produces a magnetic field opposing this decrease. To oppose the decrease in outward flux, the induced current must create an outward magnetic field, reinforcing the weakening field. Using the Right-Hand Rule, the current must flow in a direction that makes the coil behave like a South pole facing the receding North pole (attraction). This induced South pole pulls the North pole of the magnet, opposing its motion away.

In both cases, the induced current opposes the motion of the magnet. Similarly, if the flux changes due to changing current in a nearby coil, the induced current creates a field that opposes the change in flux.

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Lenz's Law and Conservation of Energy

Lenz's Law is a direct consequence of the law of conservation of energy.

Consider the case where the North pole of a magnet is pushed towards a coil. According to Lenz's Law, the induced current creates a North pole facing the approaching magnet, opposing the motion. This opposition means that external work must be done to push the magnet against this repulsive force. This mechanical work done is converted into the electrical energy of the induced current (which is then dissipated as heat in the coil's resistance or converted to other forms).

If the induced current aided the motion (i.e., created an attractive force when the magnet is approaching), then the magnet would accelerate, and the induced current would increase, further increasing the force, leading to a runaway process where energy is continuously generated from nothing. This would violate the conservation of energy. Therefore, the induced current must always oppose the change that causes it.

The negative sign in Faraday's Law ($\mathcal{E} = -N \frac{d\Phi_B}{dt}$) mathematically represents Lenz's Law. The negative sign indicates that the induced EMF opposes the change in magnetic flux. If the flux is increasing ($d\Phi_B/dt > 0$), the induced EMF is negative, meaning it tries to drive current in a direction that would decrease the flux. If the flux is decreasing ($d\Phi_B/dt < 0$), the induced EMF is positive, trying to drive current in a direction that would increase the flux.

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Answer:

Motional Electromotive Force

Electromagnetic induction can occur when a conductor moves in a magnetic field, leading to what is called motional EMF. This is one way to change the magnetic flux linked with a circuit.

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Derivation of Motional EMF from Lorentz Force

Consider a straight conductor PQ of length $l$ moving with a constant velocity $\vec{v}$ perpendicular to its length in a uniform magnetic field $\vec{B}$, which is also perpendicular to both $\vec{v}$ and the conductor.

A conductor rod moving in a magnetic field, inducing motional EMF.

Let the conductor be along the y-axis, $\vec{l} = l\hat{j}$. Let the velocity be along the +x-axis, $\vec{v} = v\hat{i}$. Let the magnetic field be along the +z-axis, $\vec{B} = B\hat{k}$.

The free electrons (charge $q=-e$) in the conductor are moving along with the conductor at velocity $\vec{v}$. Thus, the electrons are moving in the magnetic field. Each electron experiences a magnetic Lorentz force:

$ \vec{F}_B = q (\vec{v} \times \vec{B}) $

Substitute the vectors:

$ \vec{F}_B = (-e) (v\hat{i} \times B\hat{k}) = -evB (\hat{i} \times \hat{k}) $

Since $\hat{i} \times \hat{k} = -\hat{j}$:

$ \vec{F}_B = -evB (-\hat{j}) = +evB \hat{j} $

The magnetic force on the electrons is directed along the +y-axis, i.e., from P towards Q along the conductor. This force causes the free electrons in the rod to move from end P towards end Q.

As electrons accumulate at end Q, end P becomes deficient in electrons (effectively positively charged). This separation of charges creates an electric field ($\vec{E}$) within the conductor, directed from the positive end P towards the negative end Q (along +y-axis). This electric field exerts an electric force ($\vec{F}_E = q\vec{E} = -e\vec{E}$) on the electrons, opposing the magnetic force.

Charge separation continues until the electric force balances the magnetic force on the drifting electrons:

$ |\vec{F}_E| = |\vec{F}_B| $

$ |q| E = |q| v B $

$ E = vB $

This electric field $E$ exists inside the moving conductor. It represents a potential difference developed across the ends of the conductor. The induced EMF ($\mathcal{E}$) is the potential difference between ends P and Q.

The potential difference $V$ across a uniform electric field $E$ over a distance $l$ is $V = El$.

$ \mathcal{E} = E l $

Substitute $E = vB$:

$ \mathcal{E} = (vB) l = Blv $

This is the formula for the magnitude of the motional EMF induced in a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$. The positive end is P and the negative end is Q (where electrons accumulate).

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Derivation of Motional EMF from Change in Magnetic Flux

Consider a rectangular conductor loop PQRS placed in a magnetic field $\vec{B}$ directed into the page. Let the side QR be a movable conductor of length $l$, sliding on the fixed parallel rails PS and QR. Let the loop be in the x-y plane, and the magnetic field $\vec{B}$ be in the +z direction (into the page). Let the rod QR move to the right with a constant velocity $\vec{v}$ (along +x).

Motional EMF induced in a loop by moving a conductor.

Let the position of the rod QR at time $t$ be $x$ from the side PS. The area of the rectangular loop PQRS is $A = lx$.

The magnetic flux through the loop at time $t$ is $\Phi_B = \vec{B} \cdot \vec{A}$. Since $\vec{B}$ is into the page and the area vector for the loop can be taken into the page, $\theta = 0^\circ$.

$ \Phi_B = B A \cos 0^\circ = B (lx) $

As the rod moves to the right, $x$ changes with time ($x = vt$, assuming $x=0$ at $t=0$). The rate of change of magnetic flux is:

$ \frac{d\Phi_B}{dt} = \frac{d}{dt}(Blx) = Bl \frac{dx}{dt} $

Since $dx/dt$ is the velocity $v$ of the rod:

$ \frac{d\Phi_B}{dt} = Blv $

According to Faraday's Law, the magnitude of the induced EMF is:

$ |\mathcal{E}| = \left|\frac{d\Phi_B}{dt}\right| = Blv $

This result matches the motional EMF derived using the Lorentz force, confirming the consistency of the two approaches.

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Direction of Induced Current (Lenz's Law for Motional EMF)

Using Lenz's Law, the induced current will flow in a direction that opposes the increase in inward flux. To oppose the increasing inward flux, the induced current must produce an outward magnetic field within the loop. Using the Right-Hand Rule, this means the induced current flows in the anti-clockwise direction (Q $\to$ R $\to$ S $\to$ P $\to$ Q).

If the rod QR is part of a closed circuit with resistance $R$, the induced current will be $I = \mathcal{E}/R = Blv/R$.

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Answer:

Energy Consideration: A Quantitative Study

Lenz's Law highlights the principle of energy conservation in electromagnetic induction. When induced current flows in a circuit due to mechanical motion (like moving a conductor in a magnetic field), a force acts on the conductor opposing its motion. Work must be done by an external agent against this opposing force. This mechanical work is converted into electrical energy (induced EMF driving the current), which is then typically dissipated as heat in the circuit's resistance or used to do work on something else (e.g., motor).

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Quantitative Analysis for Motional EMF

Let's consider the same setup as the motional EMF derivation: a rod QR of length $l$ moving with velocity $v$ along horizontal rails in a uniform magnetic field $\vec{B}$ perpendicular to the plane of motion (say, into the page). The rod and rails form a closed circuit with total resistance $R$.

We found the induced EMF: $\mathcal{E} = Blv$.

If the circuit is closed, the induced current is:

$ I = \frac{\mathcal{E}}{R} = \frac{Blv}{R} $

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Force on the Moving Conductor

Now, this current-carrying rod QR (length $l$) is in the magnetic field $\vec{B}$. It will experience a magnetic force:

$ \vec{F}_{mag} = I (\vec{l} \times \vec{B}) $

The current $I$ flows through the rod from Q to R (anti-clockwise in the loop as per Lenz's Law). The length vector $\vec{l}$ is from Q to R (along +y in our previous coordinate system). The magnetic field $\vec{B}$ is into the page (along +z).

$ \vec{F}_{mag} = I (l\hat{j} \times B\hat{k}) = IlB (\hat{j} \times \hat{k}) = IlB \hat{i} $

The force is in the +x direction. Wait, Lenz's Law says the force must OPPOSE the motion. The motion is in the +x direction. So the force should be in the -x direction. Let's recheck the direction of the induced current based on flux change.

Flux into the page ($+z$) is increasing. Induced current must create a field OUT of the page ($-z$). By right hand rule for a loop, current must be anti-clockwise (Q->R->S->P->Q). So current in rod is Q to R. $\vec{l}$ is from Q to R. $\vec{B}$ is into page. $\vec{l} \times \vec{B}$ is indeed $\hat{j} \times \hat{k} = \hat{i}$. So force is along $\hat{i}$. Ah, my assumption about the direction of velocity was +x, and the figure implies motion to the right. Let's assume motion to the right is +x.

Okay, let's redraw the diagram and re-evaluate the directions carefully. Let the rails be along x-axis, length $l$ along y-axis, motion along +x, B along +z (into page). Rod is from $(x, 0)$ to $(x, l)$. Velocity is $v \hat{i}$. $\vec{B} = B\hat{k}$. Length vector of rod is $\vec{l} = l \hat{j}$ (from Q to R).

Force on charge in rod: $\vec{F}_B = q(\vec{v} \times \vec{B})$. For electrons $q=-e$. $\vec{v} \times \vec{B} = (v\hat{i}) \times (B\hat{k}) = vB (\hat{i} \times \hat{k}) = -vB\hat{j}$. Force on electron is $-e(-vB\hat{j}) = +evB\hat{j}$. So electrons move towards $+y$ (from P to Q). Thus current is from Q to P (opposite to electron flow). Let current direction be $\vec{l}$ from Q to P, vector length $l\hat{j}$.

Current $I$ flows from Q to P. Length vector for force calculation should be in the direction of current flow, from Q to P. So $\vec{l}_{QP} = l\hat{j}$. But wait, $\vec{l}$ in $I(\vec{l} \times \vec{B})$ is the vector in the direction of current flow. So current is along +y in rod (Q to P)? No, current is along -y in rod (P to Q).

Let's reconsider. Rod PQ. P is at top, Q is at bottom. Length $l$. Moving right with $v$. B into page. Electrons in rod experience force $\vec{F}_B = (-e) (\vec{v} \times \vec{B})$. $\vec{v}$ right, $\vec{B}$ into page. $\vec{v} \times \vec{B}$ is downwards. So force on electron is upwards. Electrons gather at P, Q becomes positive. Current flows from Q to P. Length vector for current is $\vec{l}_{QP}$.

Okay, maybe the diagram labelling is reversed in the source? Let's assume the rod is QR and moves to the right. The induced EMF is $Blv$, with Q becoming positive and R becoming negative (or vice versa depending on direction of v and B). Let's assume Q is positive, R is negative. If it's a closed loop, current flows from Q through external circuit to R.

Let's use the direction from Lenz's Law directly. Motion is to the right, increasing inward flux. Induced current is anti-clockwise (Q $\to$ R $\to$ S $\to$ P $\to$ Q). So current flows from Q to R in the rod. The vector length $\vec{l}$ for the force calculation is in the direction of current flow, from Q to R. Let QR be vertical, length $l$. Let motion be horizontal, $v$. Let B be vertical, perpendicular to both. This is inconsistent with the previous setup.

Let the rod PQ (length $l$) move to the right (velocity $v$) in a field $\vec{B}$ into the page. Let P be at the top, Q at the bottom. $\vec{v}$ is right, $\vec{B}$ is into page. $\vec{v} \times \vec{B}$ is downwards. For charge $q$, force is $q(\vec{v} \times \vec{B})$. For electrons $q=-e$, force is upwards. Electrons gather at P, Q becomes positive. So induced EMF makes Q positive relative to P. Current flows from Q to P in the external circuit.

If the rod is QR of length $l$, moving right with velocity $v$, in $\vec{B}$ into the page. Let Q be at top, R at bottom. $\vec{v}$ right, $\vec{B}$ into page. $\vec{v} \times \vec{B}$ is downwards. Electrons experience force upwards, gather at Q. R becomes positive. Induced EMF makes R positive relative to Q. Current flows from R to Q in the external circuit.

Let's use the provided diagram with QR moving right. Current is anti-clockwise (Q $\to$ R $\to$ S $\to$ P $\to$ Q). So current flows from Q to R in the rod. The length vector for the force calculation is from Q to R, length $l$, direction downwards (say -y if motion is +x, B is +z). Current $I$ is flowing downwards. Length vector $\vec{l}$ for QR is $l(-\hat{j})$. Magnetic field $\vec{B}$ is $B\hat{k}$ (into page). Velocity is $v\hat{i}$.

Force on rod QR: $\vec{F}_{mag} = I \vec{l} \times \vec{B} = I (l(-\hat{j})) \times (B\hat{k}) = -IlB (\hat{j} \times \hat{k}) = -IlB \hat{i}$.

This force $\vec{F}_{mag}$ is in the $-\hat{i}$ direction, i.e., to the left, opposing the velocity $\vec{v}$ which is in the $+\hat{i}$ direction (to the right). This is consistent with Lenz's Law.

The magnitude of this opposing force is $F_{mag} = IlB$.

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Mechanical Power Input

To maintain the constant velocity $v$ of the rod, an external force $\vec{F}_{ext}$ must be applied to counteract this magnetic force. For constant velocity, the net force must be zero, so $\vec{F}_{ext} + \vec{F}_{mag} = 0$, or $\vec{F}_{ext} = -\vec{F}_{mag}$.

$ \vec{F}_{ext} = +IlB \hat{i} $

The external force has magnitude $F_{ext} = IlB$ and is directed to the right, in the direction of motion.

The rate at which the external agent does mechanical work is the mechanical power input:

$ P_{mech} = \vec{F}_{ext} \cdot \vec{v} = (IlB \hat{i}) \cdot (v \hat{i}) = IlBv $

Substituting the expression for current $I = Blv/R$:

$ P_{mech} = \left(\frac{Blv}{R}\right) l B v = \frac{B^2 l^2 v^2}{R} $

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Electrical Power Output (Power Dissipation)

The induced EMF $\mathcal{E} = Blv$ drives the current $I = Blv/R$ through the circuit with resistance $R$. The electrical energy generated is dissipated as heat in the resistor. The rate of electrical energy dissipation (electrical power) is given by:

$ P_{elect} = I^2 R $

Substitute the expression for current $I = Blv/R$:

$ P_{elect} = \left(\frac{Blv}{R}\right)^2 R = \frac{B^2 l^2 v^2}{R^2} R = \frac{B^2 l^2 v^2}{R} $

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Energy Conservation Check

Comparing the mechanical power input ($P_{mech}$) and the electrical power output (dissipated) ($P_{elect}$):

$ P_{mech} = \frac{B^2 l^2 v^2}{R} $

$ P_{elect} = \frac{B^2 l^2 v^2}{R} $

The two expressions are identical. The rate at which mechanical work is done by the external agent in moving the conductor is exactly equal to the rate at which electrical energy is dissipated as heat in the circuit.

This quantitative analysis demonstrates that the mechanical energy input is completely converted into electrical energy, which is then dissipated. This confirms the principle of energy conservation and provides a quantitative validation of Lenz's Law – the opposing force ensures that energy is conserved. The source of the induced electrical energy is the mechanical work done against the magnetic force.

This principle is fundamental to the operation of electric generators, which convert mechanical energy into electrical energy based on electromagnetic induction and the energy transfer described here.

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Answer: