Energy Consideration: A Quantitative Study with Lenz Law

Through several studies, it has been established that Lenz’ s Law maintains a consistency with the law of conservation of energy. While this was discussed in qualitative terms, let’s prove this further through a quantitative study.  Faraday’s law gives the magnitude of the induced emf which depends on rate of change. 

Read Also: Magnetism and Matter

Keyterms: Lenz’s law, Faradey’s Law, EMF, Magnetic Flux, Bar magnet, Consideration Energy, Magnetism, Energy, Gauss law

---

Lenz’s Law and Conservation of Energy

[Click Here for Sample Questions]

We all know force and energy. So, Energy Consideration is what provides the link between two, the force and the energy. Energy consideration can be used to solve Newton problems. 

In 1834, German Physicist Heinrich Friedrich Lenz introduced a rule which is known as Lenz’s Law. It gives the polarity of the induced EMF clearly and concisely. The statement of the law is-

“The polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.”

Also Read:

---

The Induced EMF is given by

[Click Here for Previous Year Questions]

The negative sign indicates this effect.

---

Lenz's Law Experiment

[Click Here for Sample Questions]

In the experiment, we see that the North Pole of a bar magnet is being pushed towards the closed coil. The magnetic flux through the coil increases as the North pole of the bar magnet moves towards the coil. Hence the current is induced in the coil in such a direction that it opposes the increase in flux.

This is possible only if the current in the coil is in a counter-clockwise direction for an observer situated on the side of the magnet. Magnetic moment associated with this current has North polarity toward the North pole of the approaching magnet. Similarly, the magnetic flux through the coil will decrease, if the North pole of the magnet is being withdrawn from the coil. To oppose this decrease in magnetic flux induced, the flow of the induced current in the coil is in the clockwise direction and the receding North pole of the bar magnet is faced by its South Pole.

As a result, the motion of the magnet is opposed by the introduced attractive force. In the above example, if we use an open circuit in place of the closed loop an emf is induced across the open ends of the circuit. Lenz’s law can determine the direction of the induced emf. 

Consider the figure, it provides an easier way to understand the direction of induced emf. To be convinced by the correctness of Lenz’s law, a little reflection should be thrown on this matter. Suppose that the induced current was in the direction opposite to the direction depicted in fig (a).

Due to induced current, the South Pole will face the approaching North pole of the magnet then, at an ever-increasing acceleration, it will be attracted towards the coil. A slight push on the magnet will initiate the process and its velocity and kinetic energy will continuously increase without expending any energy. If this can happen one could construct a perpendicular motion machine by a suitable arrangement, which does not obey the law of conservation of energy and hence can not happen.

Induced EMF

---

What is Consideration Energy?

[Click Here for Previous Year Questions]

For performing a day-to-day task we apply force on our body, we need energy for exerting our body. Here we can find a link between the force and the energy and also the thing that provides a link between these two quantities is the energy consideration.

We can prove that motional emf is correct or valid according to the conservation of energy through the concept of energy consideration in motional emf. Through mathematical derivation we are going to prove the validity, that’s why we are discussing the concept of consideration of energy individually.

Also Read:

---

Energy Consideration Physics:

[Click Here for Sample Questions]

Here we are going to discuss the energy consideration of motional emf. When we apply the principles of energy consideration in motion emf, then we will be the focus on two important concepts- Lenz’s law and the law of energy conservation.

Energy Consideration- A Quantitative Study

From the previous discussion, we see that Lenz’s law is consistent with the law of conservation of energy. The qualitative analysis was done previously. Now the same can be proved quantitatively as well. To prove the fact let us take an example-

Energy Consideration

Let us consider a rectangular loop with the sides PQ, QR, RS, and SP. The sides of the loop QR, RS, SP are fixed and the side PQ is free to move.

Let the resistance of the movable arm be “r”. Also let the resistance of the remaining arms QR, RS, SP be negligible compared to r. Therefore, we can say that the overall resistance of the rectangular loop is “r” and we can also say that the resistance of the loop does not change with the motion in the loop. We apply the voltage across the loop be £, then current in the loop is given by-

A force acts on the arm PQ if the coil is placed in the presence of a magnetic field and there is a current in the coil. The force (IxB) that acts on the coil is directed outwards which is in the direction opposite to the direction of the motion of the rod. The force acting on the arm is given by-

Here, I= the current passing through the coil

l= the length of the arm

v= the velocity of the arm in the field

B= the magnitude of the magnetic field on the system

Drift velocity is the entity that is responsible for the current in the rod, the force acting on the arm arises due to the drift velocity of the charges in the rod. If the arm PQ is pushed with constant velocity v, then we require a power to accomplish this push-

The energy that goes into the motion of the rod and is dissipated in the form of heat is given by-

The value of P_j is equal to the value of P.

Thus, to move the arm PQ we need mechanical energy, and this mechanical energy is converted into electrical energy (the induced emf) and then to thermal energy. There is a relationship between charge flow through the circuit and the change in magnetic flux. From Faraday’s law, we see that the magnitude of the induced emf is –

We know that-

And thus, from these two equations, we can write-

Read More: Inductance formula

---

Previous Year Questions 

1.  If a transformer of an audio amplifier has output impedance 8000 0 and the speaker has input impedance…...[JCECE 2005]
2. A conducting loop in the shape of a right angled isosceles triangle of height 10cm10cm is kept such that the 90^∘ vertex is…..[JEE Advance 2016]
3. A 10m long horizontal wire extends from North East to South West. It is falling with a speed of 5.0ms⁻¹……. [ JEE Main 2019]
4. If a current of 2.0A2.0A flows through the smaller loop, then the flux linked with bigger loop is…… [JEE Main 2013]
5. A coil of cross-sectional area A having n turns is placed in a uniform magnetic field B….. [JEE Main 21018]
6. A copper rod of mass m slides under gravity on two smooth parallel rails, with separation ll and set at an angle of θ with the horizontal….. [JEE Main 2018]
7. A copper wire is wound on a wooden frame, whose shape is that of an equilateral…. [JEE Main 2019]
8. A metallic rod of length ll is tied to a string of length 2l and made to rotate with angular speed…. [JEE Main 2013]
9. A square frame of side 10 cm and a long straight wire carrying current 1 A are in the plane of the paper…. [JEE Main 2014]
10. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is… [JEE Main 2019]
11. Figure shows a circular area of radius R where a uniform magnetic field….
12. In a coil of resistance 100Ω , a current is induced by changing the magnetic flux through it….. [JEE Main 2017]
13. When current in a coil changes from 5A  to 2A…. [JEE Main 2015]
14. Which radiation in sunlight, causes heating effect? 
15. X -rays are….
16. Arrange the following in decreasing order of wavelength
17. Which is having minimum wavelength...[NEET 2002]
18. The speed of radio-waves is equal to….. [JIPMER 1998]
19. Gamma rays and visible light waves rays are a,ba,b and cc respectively, then….[UPSEE 2016]

---

Things to Remember

• Energy Consideration is what provides the link between two, the force and the energy.
• Energy consideration can be used to solve Newton problems. 
• In 1834, German Physicist Heinrich Friedrich Lenz introduced a rule which is known as Lenz’s Law.
• It gives the polarity of the induced EMF clearly and concisely.
• The statement of the law is- “The polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.”
• When we apply the principles of energy consideration in motion emf, then we will be the focus on two important concepts- Lenz’s law and the law of energy conservation.