Lenz's Law: Direction of Induced Currents

Hello! Let's get started on our next lesson.

In our last session, we introduced Faraday's Law of Induction, $\mathcal{E}=-N\frac{d{\Phi }_B}{dt}$, and used it to calculate the magnitude of the induced EMF by focusing on the term $|\frac{d{\Phi }_B}{dt}|$. We noted that the negative sign contains crucial information about the direction of the induced current, which we set aside for today.

This lesson is dedicated entirely to that negative sign. We will explore Lenz's Law, the physical principle it represents. By the end of this lesson, you will be able to apply Lenz's Law systematically to determine the direction of any induced current. This is a fundamental skill for understanding how generators, motors, and transformers operate.

1. The Principle of Lenz's Law

Lenz's Law provides the "direction" part of Faraday's Law. It states:

The direction of the induced current is such that the magnetic field it creates opposes the change in magnetic flux that produced it.

The key phrase here is "opposes the change." This is a common point of confusion. The induced current does not necessarily oppose the magnetic field itself, but rather the change in the field's flux.

Let's break this down into two cases:

1. If the magnetic flux through a loop is increasing, the induced current will create a magnetic field in the opposite direction to the original field, in an attempt to "push back" against the increase.
2. If the magnetic flux through a loop is decreasing, the induced current will create a magnetic field in the same direction as the original field, in an attempt to "prop up" the diminishing flux.

In essence, the system exhibits a kind of inertia; it tries to maintain a constant magnetic flux. Your background in economics and systems thinking might bring to mind the concept of a system seeking to maintain equilibrium. When perturbed by a change in flux, it generates a response to counteract that perturbation.

2. Lenz's Law and Conservation of Energy

Why must the induced current oppose the change in flux? Lenz's Law is a direct consequence of the law of conservation of energy.

Consider what would happen if the law were reversed—if the induced current aided the change in flux.

• If you pushed a magnet towards a loop, increasing the flux, the "reversed" law would induce a current that further increased the flux.
• This stronger flux would induce a yet stronger current, which would create an even stronger flux, and so on.

This would create a positive feedback loop, generating a runaway current and an infinite amount of electrical energy from a single small push. This would be a perpetual motion machine, violating the conservation of energy.

Therefore, the universe must work according to Lenz's Law. The opposing force means you must do mechanical work to push the magnet into the coil (or pull it out). This work you perform is precisely what gets converted into the electrical energy dissipated in the loop.

The following video from Khan Academy provides an excellent and clear explanation of this fundamental connection to energy conservation.

3. A Systematic Procedure for Applying Lenz's Law

To reliably find the direction of an induced current, we can follow a four-step procedure. This method combines the concept of flux change with the right-hand rule you've used before.

Step 1: Determine the direction of the external magnetic field.
Identify the direction of the magnetic field ${\vec{B}}_{ext}$ passing through the conducting loop.

Step 2: Determine how the magnetic flux is changing.
Is the magnitude of the flux, ${\Phi }_B$, increasing, decreasing, or staying constant? Remember that flux can change due to a change in $B$, area $A$, or angle $\theta$.

Step 3: Determine the direction of the induced magnetic field.
Apply Lenz's Law. The induced field ${\vec{B}}_{ind}$ must oppose the change from Step 2.

• If flux is increasing, ${\vec{B}}_{ind}$ must point opposite to ${\vec{B}}_{ext}$.
• If flux is decreasing, ${\vec{B}}_{ind}$ must point in the same direction as ${\vec{B}}_{ext}$.

Step 4: Determine the direction of the induced current.
Use the right-hand rule in reverse. Point the thumb of your right hand in the direction of the induced field, ${\vec{B}}_{ind}$, that you found in Step 3. Your fingers will curl around your thumb in the direction of the required induced current, $I_{ind}$.

4. Worked Examples

This procedure is best understood through examples. The following video walks through several common scenarios, applying these steps methodically.

After watching the examples, let's consolidate your understanding with a problem.

Test your understanding!

Consider a circular conducting loop in the plane of your screen. A uniform magnetic field is directed out of the screen through the loop. The strength of this magnetic field is decreasing over time.

What is the direction of the induced current in the loop? Clockwise or counter-clockwise?

Show answer

Let's apply the four-step procedure:

1. External Field Direction: The external field ${\vec{B}}_{ext}$ is directed out of the screen.
2. Flux Change: The field strength is decreasing, so the magnetic flux through the loop is decreasing.
3. Induced Field Direction: To oppose the decrease, the induced field ${\vec{B}}_{ind}$ must "help" the external field. Therefore, ${\vec{B}}_{ind}$ must also be directed out of the screen.
4. Induced Current Direction: Using the right-hand rule, point your thumb out of the screen (in the direction of ${\vec{B}}_{ind}$). Your fingers curl in the counter-clockwise direction.

Therefore, the induced current is counter-clockwise.

5. Further Examples

The MIT course notes we've used previously contain a number of solved problems that are excellent for reinforcing this procedure. Reviewing them will show you how the same physical law applies in different contexts.

Conclusion

In this lesson, we demystified the negative sign in Faraday's Law by introducing Lenz's Law. We have seen that it is not just a mathematical convention, but a fundamental consequence of the conservation of energy.

Key Takeaways:

• Lenz's Law states that an induced current will flow in a direction that creates a magnetic field to oppose the change in magnetic flux.
• This opposition is necessary to conserve energy; the work done against the opposing magnetic force is converted into electrical energy.
• We established a reliable four-step procedure to determine the direction of any induced current: (1) Find ${\vec{B}}_{ext}$, (2) Determine $\Delta {\Phi }_B$, (3) Find ${\vec{B}}_{ind}$, and (4) Use the right-hand rule to find $I_{ind}$.

Preview of the Next Lesson:
We will now circle back to motional EMF, the first type of induction we studied. In the next lesson, we will re-examine motional EMF as a specific application of Faraday's and Lenz's laws, showing how the flux-change perspective perfectly aligns with the Lorentz force picture we started with. This will help unify these concepts.