KAHIBARO
Table of Contents
The Direction of Induced Effects
Lenz's law tells us the direction of an induced current or induced electromotive force. It says that when the magnetic flux through a loop changes, the induced current appears in such a direction that the magnetic field created by that current opposes the change in flux.
This is a direction rule. Faraday's law tells us that a changing magnetic flux induces an emf. Lenz's law tells us which way that induced emf acts.
Lenz's law: The induced current always flows so that its own magnetic field opposes the change in magnetic flux that produced it.
This idea is written mathematically in Faraday's law by the minus sign,
$$
\mathcal{E} = -\frac{d\Phi_B}{dt}
$$
The minus sign is the mathematical statement of Lenz's law.
What "Opposes the Change" Means
It is very important to understand that the induced current does not always oppose the magnetic field itself. It opposes the change in magnetic flux.
This difference matters. If the external magnetic flux through a loop is increasing, the induced current creates a magnetic field that tries to reduce that increase. If the external magnetic flux is decreasing, the induced current creates a magnetic field that tries to keep the flux from falling.
A simple summary is shown below.
| Change in external flux | Induced magnetic field |
|---|---|
| Flux increasing into the page | Out of the page |
| Flux decreasing into the page | Into the page |
| Flux increasing out of the page | Into the page |
| Flux decreasing out of the page | Out of the page |
A Simple Loop Example
Imagine a circular conducting loop. A magnetic field points into the page, and the field strength increases. Because the flux into the page is increasing, the induced current must create a magnetic field out of the page to oppose that increase.
Using the right hand rule, a magnetic field out of the page is produced by a counterclockwise current.
If instead the magnetic field into the page were decreasing, the loop would try to preserve it. Then the induced current would create a field into the page, which requires a clockwise current.
Using the Right Hand Rule with Lenz's Law
In many problems, the method is:
First decide whether the magnetic flux is increasing or decreasing.
Next decide what direction of induced magnetic field would oppose that change.
Finally use the right hand rule to find the current direction that produces that induced field.
For a loop, the right hand rule works like this. Curl the fingers of your right hand in the direction of current. Your thumb points in the direction of the magnetic field produced through the loop.
Problem solving rule: Determine the change in flux first, then choose the induced field that opposes that change, then use the right hand rule to find the current direction.
Lenz's Law and Energy Conservation
Lenz's law is closely connected to conservation of energy. If the induced current helped the change instead of opposing it, the system could produce larger and larger currents without any input of work. That would violate energy conservation.
For example, if you push a magnet toward a conducting loop, the induced current produces a magnetic effect that resists the motion. You must do work to keep the magnet moving. That work is converted into electrical energy and sometimes thermal energy in the conductor.
So Lenz's law is not just a sign convention. It is required by physics.
Physical meaning: Lenz's law ensures that induced electrical effects do not create energy from nothing.
Moving Magnet and Coil
Consider a bar magnet approaching a coil. Suppose the north pole faces the coil. As the magnet gets closer, the magnetic flux through the coil increases. The coil responds by producing its own magnetic field that opposes this increase.
That means the face of the coil nearest the magnet acts like a north pole, repelling the approaching north pole. If the magnet moves away, the flux decreases, and the coil tries to keep the flux from decreasing. Then the near face acts like a south pole, attracting the retreating north pole.
Conducting Rod Example
Lenz's law also applies when a conductor moves in a magnetic field. If a rod slides on rails in a magnetic field, the motion changes the area of the loop and therefore changes the magnetic flux. The induced current then flows in a direction such that the magnetic force on the rod opposes the rod's motion.
If the rod moves to the right and the flux into the page increases, the induced current must create a field out of the page. That determines the current direction in the loop. Once current exists, the rod feels a magnetic force to the left, opposing the motion.
This is another direct expression of Lenz's law.
Common Mistakes
A frequent mistake is to ask, "What direction opposes the magnetic field?" That is not the correct question. The correct question is, "What direction opposes the change in magnetic flux?"
Another common mistake is to forget that flux depends on more than field strength. Flux can change because the magnetic field changes, because the loop area changes, or because the angle changes. In all cases, Lenz's law responds to the change in flux.
Practical Importance
Lenz's law appears in many devices. In electrical generators, the induced current resists the turning motion, so mechanical work is needed. In magnetic braking, induced currents in a moving conductor create forces that oppose motion. In transformers and inductors, the induced effects always act to oppose changes in current or flux.
So Lenz's law is one of the main ideas that makes electromagnetic induction physically consistent and useful.
Essential statement: The induced current acts in whatever direction is necessary to oppose the change in magnetic flux through the circuit.