Lenz's Law: Videos & Practice Problems

Faraday's law states that a changing magnetic flux induces an electromotive force (emf). Lenz's law determines the direction of this induced current, which opposes the change in magnetic flux. For example, if a bar magnet moves downward, the induced magnetic field points upward, resulting in a counterclockwise current. Conversely, if a loop exits a magnetic field, the induced current is clockwise. The relationship can be expressed as $\epsilon =-n\times \frac{\Delta }{\Phi }/\Delta t$.

Lenz's Law

In the study of electromagnetism, understanding the relationship between changing magnetic flux and induced electromotive force (EMF) is crucial. According to Faraday's law, a change in magnetic flux through a surface generates an EMF, which can be related to the induced current using Ohm's law. However, determining the direction of this induced current requires the application of Lenz's law. This law states that the direction of the induced current will create a magnetic field that opposes the change in magnetic flux. The key concept here is that the induced current acts to counteract the change occurring in the system.

To visualize the direction of the induced current, the right-hand rule is employed. When applying this rule, the thumb points in the direction of the magnetic field, while the fingers curl in the direction of the induced current. This method is essential for analyzing various scenarios involving magnetic fields and induced currents.

Mathematically, Faraday's law can be expressed as:

$$\epsilon = -n \frac{\Delta \Phi}{\Delta t}$$

where \(\epsilon\) is the induced EMF, \(n\) is the number of loops, and \(\Delta \Phi\) represents the change in magnetic flux over time \(\Delta t\). The negative sign in this equation reflects Lenz's law, indicating that the induced EMF acts in the opposite direction to the change in magnetic flux.

For example, consider a bar magnet moving downward through a loop. As the magnet approaches, the magnetic field through the loop increases, leading to a positive change in magnetic flux. According to Lenz's law, the induced current will flow in a direction that creates a magnetic field opposing this increase, resulting in an upward-directed induced magnetic field. Using the right-hand rule, this configuration yields a counterclockwise current when viewed from above.

In another scenario, if a loop is moving out of a magnetic field, the area through which the magnetic field lines pass decreases, resulting in a negative change in magnetic flux. Here, the induced magnetic field will act to counteract this decrease, reinforcing the existing downward magnetic field. Again, applying the right-hand rule reveals that the induced current will flow in a clockwise direction when viewed from above.

These principles can be summarized as follows: when the magnetic field is increasing, the induced magnetic field will oppose that increase, and when the magnetic field is decreasing, the induced magnetic field will act to restore the original state. This consistent behavior of induced currents is fundamental in understanding electromagnetic induction and its applications in various technologies.

An outer ring is connected to a variable voltage source. If the battery's voltage is continuously INCREASING, what is the direction of the induced current in the inner ring, centered inside of the outer ring?

• A. Clockwise
• B. Counterclockwise
• C. Into the page
• D. Out of the page

Answer: (A)

Lenz's Law for a Long Straight Wire

In this scenario, we analyze a long straight wire carrying a constantly increasing current in the positive y-direction, alongside a square loop positioned in the xy-plane. To determine the direction of the induced current in the loop, we apply Lenz's Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux through the loop.

First, we establish the direction of the magnetic field generated by the current in the wire. The magnetic field (B) around a straight current-carrying wire can be calculated using the formula:

$$ B = \frac{\mu_0 I}{2 \pi r} $$

where \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( r \) is the distance from the wire. Using the right-hand rule, we point our thumb in the direction of the current (upward along the y-axis), and our fingers curl around the wire, indicating that the magnetic field lines circle the wire in a clockwise direction when viewed from above. Therefore, at the location of the square loop, the magnetic field points downwards.

Next, we assess how the magnetic flux (\( \Phi_B \)) through the loop is changing. The magnetic flux is given by:

$$ \Phi_B = B \cdot A \cdot \cos(\theta) $$

where \( A \) is the area of the loop and \( \theta \) is the angle between the magnetic field and the normal to the surface of the loop. In this case, the area remains constant, and the angle does not change, but as the current increases, the magnetic field strength (B) also increases. Thus, the change in magnetic flux is primarily due to the increasing magnetic field.

According to Lenz's Law, since the magnetic field through the loop is increasing downwards, the induced current will flow in a direction that creates an opposing magnetic field. This means the induced magnetic field must point upwards. To find the direction of the induced current, we again use the right-hand rule, this time with our thumb pointing in the direction of the induced magnetic field (upwards). Our fingers will then curl in the direction of the induced current.

From this perspective, the induced current flows in a counterclockwise direction around the loop. Therefore, the final conclusion is that the induced current in the square loop is counterclockwise, effectively opposing the increase in downward magnetic flux as dictated by Lenz's Law.

Here’s what students ask on this topic:

What is Lenz's Law and how does it relate to Faraday's Law?

Lenz's Law states that the direction of the induced current in a conductor due to a changing magnetic flux will be such that it creates a magnetic field opposing the change in flux. This is a direct consequence of the conservation of energy. Faraday's Law quantifies the induced electromotive force (emf) as $\epsilon =\; -n\frac{\mathrm{\Delta \Phi }}{\mathrm{\Delta t}}$, where $\epsilon$ is the emf, $n$ is the number of turns, and $\mathrm{\Delta \Phi }$ is the change in magnetic flux. The negative sign in Faraday's Law is a representation of Lenz's Law, indicating the opposition to the change in magnetic flux.

How do you determine the direction of the induced current using Lenz's Law?

To determine the direction of the induced current using Lenz's Law, follow these steps: First, identify the direction of the changing magnetic flux. If the magnetic flux is increasing, the induced current will create a magnetic field opposing this increase. If the flux is decreasing, the induced current will create a magnetic field to oppose the decrease. Use the right-hand rule: point your thumb in the direction of the induced magnetic field, and your fingers will curl in the direction of the induced current. This method ensures that the induced current always opposes the change in magnetic flux.

What is the right-hand rule in the context of Lenz's Law?

The right-hand rule is a mnemonic used to determine the direction of the induced current in a loop due to a changing magnetic field. To apply it, point your thumb in the direction of the induced magnetic field (which opposes the change in flux). Your fingers will naturally curl in the direction of the induced current. This rule helps visualize how the induced current creates a magnetic field that opposes the original change in magnetic flux, as stated by Lenz's Law.

Can you provide an example of Lenz's Law in action?

Consider a bar magnet moving downward through a loop of wire. As the magnet approaches, the magnetic flux through the loop increases. According to Lenz's Law, the induced current will create a magnetic field opposing this increase. Using the right-hand rule, if the induced magnetic field points upward, the induced current will flow counterclockwise when viewed from above. This counterclockwise current generates a magnetic field that opposes the downward motion of the bar magnet, demonstrating Lenz's Law in action.

Why is the negative sign important in Faraday's Law?

The negative sign in Faraday's Law, $\epsilon =\; -n\frac{\mathrm{\Delta \Phi }}{\mathrm{\Delta t}}$, is crucial because it indicates the direction of the induced emf and current. This negative sign embodies Lenz's Law, which states that the induced current will always oppose the change in magnetic flux. Without this negative sign, the equation would only provide the magnitude of the induced emf, not its direction. The negative sign ensures that the induced current creates a magnetic field that counteracts the change in the original magnetic flux, maintaining the conservation of energy.