Problem 5 - University Physics with Modern Physics

Question

A circular loop of wire with a radius of 12.0 cm and oriented in the horizontal \(xy\)-plane is located in a region of uniform magnetic field. A field of 1.5 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 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) The average induced emf is approximately 106.5 V. (b) The induced current is counterclockwise when viewed from above.

Step 1: Understand Faraday's Law of Induction

Faraday's Law states that the average electromotive force (emf) induced in a loop is equal to the rate of change of the magnetic flux through the loop. It can be expressed as: \[ \text{emf} = - \frac{\Delta \Phi}{\Delta t} \]where \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time interval.

Step 2: Calculate Initial Magnetic Flux

The magnetic flux \( \Phi \) through the loop is given by:\[ \Phi = B \cdot A = B \cdot \pi r^2 \]where \( B \) is the magnetic field, and \( A \) is the area of the loop. For a radius \( r = 0.12 \) m and \( B = 1.5 \) T, we compute:\[ \Phi = 1.5 \times \pi \times (0.12)^2 \]

Step 3: Perform the Flux Calculation

Now, actually calculate the initial magnetic flux:\[ \Phi = 1.5 \times \pi \times 0.0144 = 0.0679 \pi \]This evaluates approximately to \( \Phi \approx 0.213 \) Wb.

Step 4: Determine Change in Flux

When the loop is removed from the magnetic field, the final magnetic flux is zero. Therefore, the change in flux is:\[ \Delta \Phi = 0 - 0.213 = -0.213 \text{ Wb} \]

Step 5: Calculate the Average Induced EMF

Using Faraday's Law:\[ \text{emf} = -\frac{\Delta \Phi}{\Delta t} = -\frac{-0.213}{0.002} \]\[ \text{emf} \approx 106.5 \text{ V} \]

Step 6: Determine Direction of Induced Current

According to Lenz's Law, the direction of the induced current is to oppose the change in magnetic flux. Since the wire loop was initially in the field and the field is removed, the induced current will have a magnetic moment that opposes the loss of upward field. Thus, when viewed from above, the current is induced in a direction that opposes the field's decrease - which is counterclockwise.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle of physics describing how an electric current is generated by changing magnetic fields. It was first discovered by Michael Faraday. This process is the foundation of many technological applications, such as electric generators and transformers.

The key idea is that a moving or changing magnetic field can produce an electric current in a conductor. This action relies on the relative motion between the magnetic field and the conductor, thus creating an electromotive force (EMF).

It is important to understand that it's not the strength of the magnetic field alone that causes induction, but how it changes over time.

Magnetic Flux

Magnetic flux (\( \Phi \)) is a measure of the number of magnetic field lines passing through a given area. You can think of it as the total magnetic field "threading" through a particular space. When dealing with a loop of wire, the magnetic flux is determined using the formula:\[ \Phi = B \times A \]where \( B \) is the magnetic field strength and \( A \) is the area of the loop.

In the exercise, the magnetic flux was calculated as a product of the magnetic field and the area of the circular wire loop. The more lines of magnetic field that pass through the loop, the greater the magnetic flux.

Lenz's Law

Lenz's Law gives direction to the induced current and EMF. It states that an induced current will flow in a direction that opposes the change that caused it. This concept can be a bit tricky at first, but it is a natural consequence of the law of conservation of energy.

In the given exercise, as the magnetic flux decreases to zero when the loop is removed from the magnetic field, an induced current circulates in the loop to maintain the original flux direction. This opposing action ensures that energy is conserved in the system.

Induced EMF

The induced electromotive force (EMF) is essentially the voltage generated by the process of electromagnetic induction. According to Faraday's law, the magnitude of the induced EMF in a closed loop is equal to the rate of change of magnetic flux through the loop:\[ \text{emf} = - \frac{\Delta \Phi}{\Delta t} \]

The negative sign indicates the direction of the induced EMF, as explained by Lenz's Law. In simple terms, faster changes in magnetic flux induce a stronger EMF.

Circular Wire Loop

A circular wire loop is a simple yet effective setup for observing electromagnetic induction. The symmetry of the circle allows for straightforward calculations of area, making it easier to apply equations related to flux and EMF.

In our problem, a circular loop with a given radius was used to determine the magnetic flux. The circular configuration ensures that calculations related to changes in the magnetic field across the loop are uniform and the resulting induced EMF can be accurately quantified.

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. In this exercise, the magnetic field is uniformly directed along the z-axis at 1.5 T. The uniformity and orientation of the field are critical as they ensure consistent induction across the loop.

Magnetic fields are represented by field lines. The direction of these lines goes from the north to the south pole outside the magnet and vice versa inside it.

The uniform magnetic field in the exercise plays a crucial role as it builds a foundation for calculating magnetic flux through the loop.

Direction of Current

The direction of the induced current in a loop can be determined using Lenz's Law. In this context, we had a loop that was initially experiencing a vertical magnetic field which was then removed. The induced current resisted this change, trying to "hold onto" the magnetic field.

Therefore, when viewed from above, the induced current flows counterclockwise, creating a field in the opposite direction to the change – attempting to maintain the original upward field.

• This concept is crucial for understanding how electric generators work, where the direction of rotation is key to harnessing current effectively.