Problem 9 - University Physics with Modern Physics

Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circunference of \(165.0 \mathrm{cm},\) but its circunference is decreasing at a constant rate of 12.0 \(\mathrm{cm} / \mathrm{s}\) due to a tangential pull on the wire. The loop is in a constant, viniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 \(\mathrm{T}\) . (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Step-by-Step Solution

Answer

(a) The induced emf is 5.454 mV. (b) The induced current is counterclockwise.

Step 1: Understanding the Problem

We have a circular loop with an initial circumference of 165.0 cm that is decreasing at 12.0 cm/s. We're in a uniform magnetic field of 0.500 T perpendicular to the loop. We need to find the induced electromotive force (emf) after 9 seconds and the direction of the induced current.

Step 2: Calculate the Circumference After 9 Seconds

The circumference decreases by 12.0 cm/s, so in 9 seconds, it decreases by \(12.0 \times 9 = 108.0\) cm. The new circumference is \(165.0 - 108.0 = 57.0\) cm.

Step 3: Calculate the New Radius

The new circumference is 57.0 cm. Using the formula for circumference, \(C = 2\pi r\), solve for the radius: \(r = \frac{57.0}{2\pi}\approx 9.07\) cm.

Step 4: Calculate the Area of the Loop

The area of the loop is given by \(A = \pi r^2\). Substituting \(r\approx 9.07\) cm: \(A = \pi \times (9.07)^2\approx 259.2\) square cm or \(0.02592\) square meters.

Step 5: Calculate the Rate of Change of Area

As the loop's circumference decreases, so does its area. Using the relationship \(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\), and noting that \(\frac{dr}{dt} = -\frac{12.0}{2\pi} \approx -1.91\) cm/s from Step 3, we find \(\frac{dA}{dt} = 2\pi \times 9.07 \times -1.91 \approx -109.08\) cm/s or \(-0.010908\) m/s.

Step 6: Apply Faraday's Law of Induction

Faraday's Law states \(\mathcal{E} = -B \frac{dA}{dt}\). Substituting \(B = 0.500\) T and \(\frac{dA}{dt} = -0.010908\ m^2/s\), we get \(\mathcal{E} = -0.500 \times (-0.010908) = 0.005454\) V or \(5.454\) mV.

Step 7: Determine the Direction of Induced Current

Using Lenz's Law, the induced current will oppose the change in magnetic flux. Since the loop is getting smaller, the flux is decreasing. To oppose this, the induced current will create a magnetic field in the same direction as the applied field. Viewing along the magnetic field direction, the current is counterclockwise.

Key Concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle of electromagnetism discovered by Michael Faraday. It describes how an electromotive force (EMF) is generated in a loop when there is a change in magnetic flux through the loop. This concept is central in understanding how electrical currents can be induced by changing magnetic fields. In this specific exercise with the shrinking loop of iron wire, Faraday's Law allows us to calculate the induced EMF.

The formula for Faraday's Law is:
\[\mathcal{E} = -\frac{d\Phi}{dt}\]where \(\mathcal{E}\) is the induced EMF and \(\frac{d\Phi}{dt}\) is the rate of change of magnetic flux \(\Phi\). In our problem, the shrinking loop reduces its area, hence changing the flux through the loop. The negative sign in the formula indicates the direction of the induced EMF. It reflects Lenz's Law by showing that the induced EMF works in opposition to the change in magnetic flux. The exercise uses Faraday's Law to find that the induced EMF in the shrinking loop is approximately 5.454 mV.

Lenz's Law

Lenz's Law is an extension of Faraday's Law that provides insight into the direction of induced currents. It states that the direction of the induced current will be such that it opposes the cause of its creation. In simpler terms, Lenz’s Law helps to determine how the induced current acts to counter the change in magnetic flux that produced it.

In this exercise, as the loop's circumference decreases, the magnetic flux through the loop decreases too. According to Lenz's Law, the induced current will circulate in a direction that opposes this reduction in flux. This means the induced current generates a magnetic field that tries to maintain the original flux through the loop.

• The loop generates a counterclockwise current when viewed along the direction of the magnetic field.
• This results in a magnetic field that adds to the original external field, thereby opposing the loss of magnetic flux.

Lenz’s Law is crucial in predicting how systems react to maintain equilibrium, providing a deeper understanding of electromagnetic interactions.

Magnetic Flux

Magnetic Flux represents the total magnetic field passing through a given area. It is a measurement of the strength of the magnetic field interacting with the loop and is defined as:
\[\Phi = B \times A \times \cos(\theta)\]where \(\Phi\) is the magnetic flux, \(B\) is the magnetic field strength, \(A\) is the area through which the field lines pass, and \(\theta\) is the angle between the magnetic field and the normal (perpendicular) to the area.

In the given exercise, the loop is in a uniform magnetic field of 0.500 T which is perpendicular to the loop, so \(\cos(\theta)\) is 1. As the loop shrinks, the area decreases, thereby reducing magnetic flux. This change in magnetic flux is what leads to the induction of an EMF according to Faraday's Law.

Understanding Magnetic Flux provides clarity on how changes in a magnetic field directly influence the creation of electric currents, drawing a foundational link in the field of electromagnetism.