Problem 5 - University Physics with Modern Physics

Question

A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) and oriented in the horizontal \(x y\) -plane is located in a region of uniform magnetic field. A field of 1.5 \(\mathrm{T}\) is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 \(\mathrm{ms}\) , find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

Step-by-Step Solution

Answer

(a) 34.05 V; (b) Counterclockwise.

Step 1: Calculate the Initial Magnetic Flux

The initial magnetic flux \( \Phi_i \) through the loop can be calculated using the formula: \[ \Phi_i = B \times A = B \times \pi r^2 \] where \( B = 1.5 \mathrm{T} \) and \( r = 0.12 \mathrm{m} \). Substitute the values: \[ \Phi_i = 1.5 \times \pi \times (0.12)^2 \approx 0.0681 \mathrm{Wb} \] (Weber).

Step 2: Determine the Final Magnetic Flux

When the loop is removed from the magnetic field, the final magnetic flux \( \Phi_f \) is 0 since there is no magnetic field acting on the loop. Hence, \( \Phi_f = 0 \mathrm{Wb} \).

Step 3: Calculate the Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is given by \( \Delta \Phi = \Phi_f - \Phi_i \). Substitute the values: \[ \Delta \Phi = 0 - 0.0681 = -0.0681 \mathrm{Wb} \].

Step 4: Calculate the Average Induced EMF

Using Faraday's Law of Induction, the average induced EMF \( \mathcal{E} \) is given by: \[ \mathcal{E} = -\frac{\Delta \Phi}{\Delta t} \] Here, \( \Delta t = 2.0 \times 10^{-3} \mathrm{s} \). Substitute the values and calculate: \[ \mathcal{E} = -\frac{-0.0681}{2.0 \times 10^{-3}} \approx 34.05 \mathrm{V} \]. Thus, the average induced EMF is approximately \( 34.05 \mathrm{V} \).

Step 5: Determine the Direction of Induced Current

According to Lenz's Law, the induced current will oppose the change in flux. Since the magnetic field is directed upward and is being removed, the induced current will create a magnetic field upward to oppose this effect. Viewed from above, a counterclockwise current is needed to create an upward magnetic field. Therefore, the induced current is counterclockwise.

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle in electromagnetism. It states that a change in magnetic flux through a loop of wire will induce an electromotive force (EMF) in the wire. This change in magnetic flux can occur due to a changing magnetic field, the movement of the wire, or both. The formula representing Faraday's Law is given by:

• \( \mathcal{E} = -\frac{d\Phi}{dt} \)

Here, \( \mathcal{E} \) is the induced EMF, and \( \Phi \) is the magnetic flux. The negative sign is due to Lenz's Law, which we'll discuss later. In our exercise, this law helps us calculate the EMF when the loop is moved out of the uniform magnetic field.

Magnetic Flux

Magnetic Flux is a measure of the strength of a magnetic field passing through a given area. It essentially tells us how much magnetic field is threaded through a surface like a loop of wire. The formula for calculating magnetic flux \( \Phi \) is:

• \( \Phi = B \times A \times \cos(\theta) \)

Here \( B \) is the magnetic field strength, \( A \) is the area of the loop, and \( \theta \) is the angle between the magnetic field lines and the normal to the surface. In our exercise, since the magnetic field is perpendicular to the loop's plane, \( \theta \) is 0, making \( \cos(\theta) \) equal to 1. Thus, \( \Phi = B \times \pi r^2 \), where \( r \) is the radius of the loop.

Lenz's Law

Lenz's Law is crucial for determining the direction of the induced current. This law states that any induced EMF will act in a direction that opposes the change in magnetic flux that produced it. Think of it as nature's version of a pushback – when a change happens, the induced current will try to resist that change. This law complements Faraday's Law, represented by the negative sign in the formula \( \mathcal{E} = -\frac{d\Phi}{dt} \). In our problem, as the loop exits the magnetic field, the induced current will attempt to maintain the magnetic flux by generating a magnetic field in the same direction as the original.

Induced EMF

Induced EMF or electromotive force is essentially the voltage generated by changing the magnetic environment of a coil of wire. In our problem, we calculated it using Faraday's Law. By finding the change in magnetic flux and dividing by the time interval, we can find the average EMF:

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

Where \( \Delta \Phi \) is the change in magnetic flux, calculated as \( \Phi_f - \Phi_i \). In this exercise, moving the loop out of the magnetic field reduced the magnetic flux to zero, resulting in a calculated induced EMF of approximately 34.05 volts.

Direction of Induced Current

The direction of induced current is a direct consequence of Lenz's Law. As we determined, the induced current opposes the change in flux resulting from moving out of the magnetic field. In practice, for a loop viewed from above, the presence and removal of an upward-directed magnetic field means that the induced current will need to reinforce the original magnetic field direction.

• To reinforce an upward magnetic field, the induced current flows counterclockwise when observed from above.

Understanding the process of determining this direction requires visualizing the magnetic field lines and how currents interact with magnetic fields, aligning with the right-hand rule for determining direction.