Lenz's Law

"Edited by Alex Casado (Spring 2019)"

The Main Idea

Lenz's Law is applies the law of conservation of energy to Faraday's Law, which states that any change in the magnetic field will cause an induced current. Lenz's law accounts for the direction of the new current - with the change coming from either the strength of the magnetic field, the direction of the magnetic field, the position of a circuit, the shape of a circuit, or the orientation of a circuit. To keep the magnetic flux of a loop of wires constant, there is an induced magnetic field. Change in magnetic flux results in an equal and opposite change in the loop which is why we use negative [math]\displaystyle{ {\frac{dB}{dt}} }[/math]

A Mathematical Model

Lenz's Law is mathematically modeled as part of Faraday's Law. The negative sign in the equation represents the opposing induced field. [math]\displaystyle{ \epsilon = -{\frac{d\phi}{dt}} }[/math] where [math]\displaystyle{ \epsilon }[/math] is the emf of the system and [math]\displaystyle{ d\phi }[/math] is the change in the magnetic field.

A Computational Model

https://www.youtube.com/watch?v=xxZenoBs2Pg This links to a video by Khan Academy that explains how Lenz's Law works through an example.

https://www.youtube.com/watch?v=Vs3afgStVy4 This links to a video by Grand Illusions that displays Lenz's Law using a magnetic falling down a tube.

Examples

Example Using Right Hand Rule

Use the right hand rule to find the non-coulombic electric field in the given situations.

Solution

To use the right hand rule: place your thumb in the direction of the [math]\displaystyle{ -dB }[/math], then curl your fingers. The direction in which your fingers curl is the direction of the non-coulombic electric field. The non-coulombic field is represented by the pink arrows.

Simple

A solenoid has current [math]\displaystyle{ I_1 }[/math] flowing around it increasing from 0 to 40A. A plain loop of wire is placed around the solenoid, perpendicular to the axis of the solenoid. An emf is produced, therefore producing an [math]\displaystyle{ I_2 }[/math]. Are [math]\displaystyle{ I_1 }[/math] and [math]\displaystyle{ I_2 }[/math] flowing in the same direction or opposite?

Solution The original current [math]\displaystyle{ I_1 }[/math] produced a magnetic field. In order to maintain the conservation of energy and Newton's Third Law, the magnetic field produced by [math]\displaystyle{ I_2 }[/math] must oppose this field. This is in accordance with Lenz's Law. Therefore, [math]\displaystyle{ I_2 }[/math] must oppose [math]\displaystyle{ I_1 }[/math] in direction.

Middling

An magnet is moving through a copper tube (velocity drawn). Find the direction of -dB/dt and the direction of the induced current. Remember to use the right hand rule.

Graphic created by Nicole Romer

Solution A)-y, clockwise B)+y, counterclockwise C)+y, counterclockwise D)-y, clockwise

The magnetic field in this graphic is decreasing at a rate of 5.0mT/s. What is the direction of the current in the circle of wire?

Solution The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. This is in accordance with Lenz's Law. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

Difficult

The magnetic field is decreasing at a rate of 5.0mT/s. The radius of the loop of wire is 5.0m, and the resistance is 5 ohms. What is the magnitude and direction of the current?

Solution

To find the magnitude of the current we must first use the formula [math]\displaystyle{ \epsilon = -{\frac{d\phi}{dt}} }[/math] to find the [math]\displaystyle{ \epsilon }[/math] representing the emf of the system. We know, more specifically, that [math]\displaystyle{ \epsilon = -NAcos\theta{\frac{d\phi}{dt}} }[/math]. Therefore, [math]\displaystyle{ \epsilon = -1*5.0^2*\pi*1*-5*10^{-3} }[/math] which resolves to [math]\displaystyle{ \epsilon = .392699 }[/math]. From there, to find the current we know that [math]\displaystyle{ I = {\frac{\epsilon}{R}} }[/math]. Plugging in the values we know we find, [math]\displaystyle{ I = {\frac{.392699}{5.0}} = .07854A }[/math]. That is the magnitude of the current. To find the direction we must use Lenz's Law. The current in the circle of wire will produce a magnetic field that needs to supplement the existing diminishing field. Therefore, the magnetic field produced needs to be into the page. Using the right hand rule, to produce a field going into the page the current in the circle of the wire must be in the clockwise direction.

Connectedness

This topic has many applications in the real world that are very interesting. For example, an application for Lenz's Law is to cause rotation to create energy. In an industry setting, Lenz's Law can be applied to electric generators or electric motors. When a current is induced in a generator, the direction of the induced current will flow in opposition of the magnetic field that created it, causing rotation of the generator.

Another real world application of Lenz's Law is in electromagnetic braking in vehicles. This process begins with electromagnets inducing eddy currents into the spinning rotor. Magnetic fields that oppose the initial change in magnetic flux are created from these eddy currents (Lenz's Law). This ultimately slows the rotor.

Another example of Lenz's Law in the real world is in Induction stovetops. These cooktops heat up as a result of changing magnetic fields and eddy currents operating according to Lenz's Law.

History

Henrich Friefrich Emil Lenz (1804-1865), a Russian physicist of German origin was born in Dorpat, nowadays Tartu, Estonia. Henrich studied chemistry and physics at the University of Dorpat in 1820 after his secondary education. From 1823 to 1826, he traveled with the navigator, Otto von Kotzebue on his third expedition around the world. During this journey he studied climate conditions, and properties of seawater. After his travels, he worked at the University of St. Petersburg, Russia, where he later became the Dean of Mathematics and Physics from 1840 to 1863. In the year of 1831, he started studying electromagnetism, and soon after in 1835, what is known today as Lenz's Law was created. Lenz was also know for carefully checking his work, testing any variable that might effect his results. Lenz died on February 10, 1865, just two days before his 61st birthday, after suffering a stroke, while in Rome.

See also

Since Lenz's Law and Farady's Law go hand in hand, Faraday's Law would be great supplemental information to read about. Newton's Third Law would also be a topic to read on for further understanding why Lenz's Law exists.

Further reading

Faraday's Law [1] Conservation Laws [2]

References

https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/pioneers/heinrich-friedrich-emil-lenz http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html http://regentsprep.org/regents/physics/phys08/clenslaw/ http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html http://farside.ph.utexas.edu/teaching/302l/lectures/node85.html http://www.electrical4u.com/lenz-law-of-electromagnetic-induction/