Faraday's Law: Videos & Practice Problems

Faraday's law describes electromagnetic induction, stating that the induced electromotive force (emf) is proportional to the rate of change of magnetic flux through a coil. The equation is expressed as ε = n×|ΔΦΔt|, where n is the number of turns, ΔΦ is the change in magnetic flux, and Δt is the change in time. Changes in magnetic field, area, or angle can alter magnetic flux, leading to induced currents in circuits, as demonstrated through practical examples.

Faraday's Law

Faraday's law is a fundamental principle in electromagnetism that describes how a changing magnetic field can induce an electromotive force (EMF) in a conducting loop. This phenomenon is crucial for understanding electromagnetic induction, which is the basis for many electrical devices. The key concept here is magnetic flux, which is defined as the product of the magnetic field (B), the area (A) through which the field lines pass, and the cosine of the angle (θ) between the magnetic field and the normal to the surface area. Mathematically, this is expressed as:

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

When the magnetic flux changes over time, it induces an EMF in the loop, which can lead to an induced current. Faraday's law quantifies this relationship with the equation:

$$\mathcal{E} = -n \frac{\Delta \Phi_B}{\Delta t}$$

Here, \(\mathcal{E}\) represents the induced EMF, \(n\) is the number of turns in the coil, \(\Delta \Phi_B\) is the change in magnetic flux, and \(\Delta t\) is the time interval over which the change occurs. The negative sign indicates the direction of the induced EMF, as described by Lenz's law, which states that the induced current will flow in a direction that opposes the change in magnetic flux.

To understand how magnetic flux can change, consider three variables: the magnetic field strength (B), the area of the loop (A), and the angle (θ) between the magnetic field and the area vector. Any change in these variables will affect the magnetic flux. For example, if the magnetic field strength increases while the area and angle remain constant, the magnetic flux increases, leading to a higher induced EMF. Conversely, if the area of the loop increases while the magnetic field and angle remain constant, the magnetic flux also increases.

In practical applications, such as calculating the induced EMF in a circuit, one might encounter a scenario where the magnetic field changes from 3 Tesla to 6 Tesla over a period of 5 seconds. If the loop has an area of 50 cm² (which converts to 0.005 m²), the change in magnetic flux can be calculated as:

$$\Delta \Phi_B = A \cdot (B_{final} - B_{initial})$$

Substituting the values gives:

$$\Delta \Phi_B = 0.005 \, \text{m}^2 \cdot (6 \, \text{T} - 3 \, \text{T}) = 0.015 \, \text{Wb}$$

Using Faraday's law, the induced EMF can then be calculated:

$$\mathcal{E} = -1 \cdot \frac{0.015 \, \text{Wb}}{5 \, \text{s}} = -0.003 \, \text{V}$$

Thus, the induced EMF is 0.03 volts (taking the absolute value). To find the induced current when a resistor of 2 ohms is connected, Ohm's law can be applied:

$$I = \frac{\mathcal{E}}{R} = \frac{0.03 \, \text{V}}{2 \, \Omega} = 0.015 \, \text{A}$$

This example illustrates the practical application of Faraday's law in calculating induced EMF and current in a circuit, emphasizing the importance of understanding the relationships between magnetic fields, area, and induced currents in electromagnetic systems.

A tightly-wound 200-turn rectangular loop has dimensions of 40cm by 70cm. A constant magnetic field of 3.5T points in the same direction as the normal of the loop. If the dimensions of the loop change to 20cm by 35cm over 0.5s, with the number of turns remaining the same, what is the induced EMF on the rectangular loop?

• A. 48 V
• B. 97 V
• C. 294 V
• D. 588 V
• E. 0 V

Answer: (C)

Current in a Circuit with a Changing Magnetic Field

In this scenario, we analyze a system involving two circular loops: a smaller loop with a radius of 5 cm (0.05 m) and a larger loop with a radius of 5 m. The larger loop initially carries a current of 6 A, which generates a magnetic field. When the larger loop is disconnected from its voltage source, the current decreases, leading to a change in the magnetic flux through the smaller loop.

The magnetic flux (\( \Phi \)) through the smaller loop is given by the equation:

\(\Phi = B \cdot A \cdot \cos(\theta)\)

where \( B \) is the magnetic field, \( A \) is the area of the smaller loop, and \( \theta \) is the angle between the magnetic field and the normal to the surface of the loop. In this case, since the magnetic field and the normal both point into the page, \( \cos(\theta) = 1\).

The area \( A \) of the smaller loop can be calculated as:

\( A = \pi r_s^2 = \pi (0.05)^2 \)

The magnetic field \( B \) generated by the larger loop is determined by the formula:

\( B = \frac{\mu_0 I}{2r} \)

where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)), \( I \) is the current, and \( r \) is the radius of the larger loop. As the current decreases to zero, the magnetic field also decreases, leading to a change in magnetic flux.

The change in magnetic flux (\( \Delta \Phi \)) can be expressed as:

\( \Delta \Phi = A \cdot \Delta B \)

Substituting the expressions for area and magnetic field, we find:

\( \Delta \Phi = \pi (0.05)^2 \left( B_{\text{final}} - B_{\text{initial}} \right) \)

Given that the final current is 0 A, the change in magnetic flux results in a negative value of approximately \( -5.92 \times 10^{-9} \, \text{Wb} \).

Next, we apply Faraday's law of electromagnetic induction to find the induced electromotive force (EMF). The induced EMF (\( \mathcal{E} \)) is given by:

\( \mathcal{E} = -n \frac{\Delta \Phi}{\Delta t} \)

Assuming the smaller loop has one turn (\( n = 1 \)), and using the change in magnetic flux and the time interval of 20 microseconds (\( 20 \times 10^{-6} \, \text{s} \)), we calculate:

\( \mathcal{E} = \frac{5.92 \times 10^{-9}}{20 \times 10^{-6}} \approx 2.96 \times 10^{-4} \, \text{V} \)

Finally, to find the induced current (\( I \)) in the smaller loop, we use Ohm's law:

\( I = \frac{\mathcal{E}}{R} \)

Given the resistance \( R \) of the smaller loop is 10 milliohms (0.01 ohms), we find:

\( I = \frac{2.96 \times 10^{-4}}{0.01} \approx 0.0296 \, \text{A} \)

Thus, the induced current is approximately 0.03 A. This problem illustrates the principles of electromagnetic induction, magnetic flux, and the relationship between current, resistance, and induced EMF.

A square conducting wire of side length 4 cm is in a 2 T magnetic field. It rotates such that the angle of the magnetic field to the normal of the square increases from 30° to 60° in 2 s. What is the induced current on the wire if its resistance is 5 mΩ?

• A. 0.85 A
• B. 0.12 A
• C. 1.2 A
• D. 1.39 mA

Answer: (B)

Here’s what students ask on this topic:

What is Faraday's Law of Electromagnetic Induction?

Faraday's Law of Electromagnetic Induction states that the induced electromotive force (emf) in a coil is proportional to the rate of change of magnetic flux through the coil. Mathematically, it is expressed as:

$\epsilon =n\times \frac{\left|\mathrm{\Delta \Phi }\right|}{\mathrm{\Delta t}}$

where $\epsilon$ is the induced emf, $n$ is the number of turns in the coil, $\mathrm{\Delta \Phi }$ is the change in magnetic flux, and $\mathrm{\Delta t}$ is the change in time. This law is fundamental in understanding how electric generators and transformers work.

How does changing the magnetic field affect the induced emf according to Faraday's Law?

According to Faraday's Law, changing the magnetic field through a coil induces an emf. The induced emf is directly proportional to the rate of change of the magnetic flux. If the magnetic field $B$ changes, the magnetic flux $\Phi$ (which is the product of the magnetic field, the area of the coil, and the cosine of the angle between them) also changes. This change in flux induces an emf in the coil. Faster changes in the magnetic field result in a higher induced emf.

What are the three ways to change the magnetic flux in a coil?

There are three ways to change the magnetic flux $\Phi$ in a coil:

1. Change the Magnetic Field ($B$): Varying the strength of the magnetic field passing through the coil.
2. Change the Area ($A$): Altering the area of the coil that is exposed to the magnetic field.
3. Change the Angle ($\theta$): Modifying the angle between the magnetic field and the normal to the area of the coil.

Any of these changes will result in a change in magnetic flux, thereby inducing an emf in the coil.

How is the induced current in a circuit related to the induced emf and resistance?

The induced current $I$ in a circuit is related to the induced emf $\epsilon$ and the resistance $R$ of the circuit by Ohm's Law, which is expressed as:

$\epsilon =IR$

Rearranging this equation to solve for the current, we get:

$I=\frac{\epsilon }{R}$

This means that the induced current is equal to the induced emf divided by the resistance of the circuit. Higher resistance results in a lower induced current for a given emf.

What is the significance of the number of turns in a coil in Faraday's Law?

In Faraday's Law, the number of turns $n$ in a coil is significant because it directly affects the induced emf. The law is expressed as:

$\epsilon =n\times \frac{\left|\mathrm{\Delta \Phi }\right|}{\mathrm{\Delta t}}$

Here, $n$ represents the number of turns in the coil. A higher number of turns increases the induced emf for a given rate of change of magnetic flux. This is because each turn of the coil experiences the same change in flux, and the effects are cumulative.