Lenz’s Law of Electromagnetic Induction

Understanding how electrical energy works when magnetic fields change is made easier by the intriguing electromagnetism concept known as Lenz law. This rule, which bears the name of the eminent scientist Emil Lenz, describes the kinetics of electromagnetic interactions.

1.0What is Lenz Law of Electromagnetic Induction

Lenz law of electromagnetic induction states that when a magnetic field changes around a conductor, it creates an induced current that always opposes the change that produced it. 

The Core Principle:

Electric current is produced when a coil or conductor is in close proximity to a magnetic field. However, this induced current produces a magnetic field of its own that attempts to obstruct the motion of the original magnetic field.

2.0Mathematical Representation

The Lenz law formula can be expressed mathematically as:

ε = -N * (dΦ/dt)

Where:

• ε represents the induced electromotive force (EMF)
• N is the number of turns in the coil
• dΦ/dt represents the rate of change of magnetic flux
• The negative sign indicates the opposing nature of the induced current

3.0Experimental Demonstrations

• First Experiment: Understanding Magnetic Flux

In the first experiment, researchers noticed that magnetic field lines are created when electricity passes through a coil. The magnetic flux rises in tandem with the current. This rise is always opposed by the direction of the generated current.

• Second Experiment: Magnetic Field Interactions

An induced current is created when a current-carrying coil coiled on an iron rod is moved. The velocity and the strength of the magnetic field determine which way this stream flows.

• Third Experiment: Coil and Magnetic Flux

As a coil is drawn into a magnetic field, its area within the magnetic field diminishes. The generated current acts against this motion exhibiting the basic concept of Lenz's law.

4.0Conservation of Energy

Lenz law and conservation of energy are intimately connected. The law ensures that the extra work done against the opposing force is converted into electrical energy. This perfectly illustrates the principle of energy conservation - no energy is created or destroyed, just transformed.

5.0Practical Implications

Difference from Faraday's Law

While Lenz's law focuses on the direction of induced current, Faraday's law concentrates on the electromagnetic force produced. Together, they provide a comprehensive understanding of the lenz law of electromagnetic induction.

6.0Solved Problems: Lenz Law Examples

Problem 1: A coil with 100 turns and a resistance of 4 ohms is placed in a uniform magnetic field. The magnetic field is increasing at a rate of dB/dt = 2 T/s. If the area of the coil is 0.01 m², find the induced current in the coil.

Solution: By using Faraday’s law: 

$ϵ=-N\frac{d{\varphi }_B}{dt}$

Here,

${\varphi }_B=B.A$

The rate of change of magnetic flux is 

$\frac{d{\varphi }_B}{dt}=A.\frac{dB}{dt}$

$\frac{d{\varphi }_B}{dt}$$=0.012=0.02Wb/s$

$=-1000.02=-2V$

The negative sign here shows that the induced EMF opposes the change in magnetic flux. 

Problem 2: Consider a solenoid of length 0.5 m, radius 0.05 m, and 500 turns, carrying a current that changes at a rate of dI/dt = 0.02 A/s. A small loop with a radius of 0.05 m is placed inside the solenoid, perpendicular to its axis. Find the induced emf in the loop and the direction of the current using Lenz’s Law.

Solution: The magnetic field inside a solenoid is given by:

$B={\mu }_{0}nI$

Here, ${\mu }_0=4\pi \times 10^{-7}Tm/A$

n is the number of turns per unit length, and I represents the current flowing.

$n=\frac{500}{0.5}=1000turns/m$

Rate of change of magnetic field: 

$\frac{dB}{dt}={\mu }_{0}n\frac{dI}{dt}=(4\pi \times 10^{-7})\times 1000\times 0.02=2.51\times 10^{-5}T/s$

Induced emf in the loop: 

$\epsilon =-A\frac{dB}{dt}$

A = area of the loop = r2=3.140.052=7.8510-3m2

$\epsilon =-(7.85\times 10^{-3})\times (2.51\times 10^{-5}=-1.97\times 10^{-7}V$

Direction of the induced current: Since the current in the solenoid is increasing, the magnetic flux through the loop will also increase. Lenz's Law states that the induced current will flow in such a direction as to oppose this increasing magnetic flux. Therefore, the current induced will create a magnetic field opposing the increasing field inside the solenoid.

Problem 3: A coil with 200 turns and a resistance of 5 ohms is placed in a magnetic field. If the magnetic field changes at the rate of 0.1 T and the area of the coil is 0.01 m², find the induced current in the coil.

Solution: Induced emf (by the faraday’s law): 

$\epsilon =-N\frac{d{\varphi }_B}{dt}=-N.A.\frac{dB}{dt}$

Given that N = 200, A = 0.01m²,

$\frac{dB}{dt}=0.1T/s$

$\epsilon =-200\times 0.01\times 0.1=-0.2V$

The negative sign shows that the induced emf is in the opposite direction of the flux. 

Induced current $I=\frac{\epsilon }{R}=\frac{-0.2}{5}=-0.04A$