Problem 9 - University Physics with Modern Physics

Question

A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 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 approximately 0.0543 V. (b) The induced current is counter-clockwise.

Step 1: Calculate the Circumference After 9 Seconds

The circumference of the loop decreases at a rate of 12.0 cm/s. After 9 seconds, the change in circumference is given by \( 12.0 \text{ cm/s} \times 9 \text{ s} = 108 \text{ cm} \). Therefore, the new circumference is \( 165.0 \text{ cm} - 108 \text{ cm} = 57.0 \text{ cm} \).

Step 2: Calculate the Radius After 9 Seconds

The formula for the circumference of a circle is \( C = 2 \pi r \). Rearranging for radius, we have \( r = \frac{C}{2\pi} \). Substituting the new circumference, \( r = \frac{57.0 \text{ cm}}{2\pi} \approx 9.07 \text{ cm} \).

Step 3: Calculate the Area of the Loop After 9 Seconds

The area of a circle is given by \( A = \pi r^2 \). Substituting the radius, \( A = \pi (9.07 \text{ cm})^2 \approx 258.85 \text{ cm}^2 \).

Step 4: Determine the Rate of Change of Area

The rate of change of circumference provides a rate of change for the radius. Since \( \frac{dC}{dt} = 12.0 \text{ cm/s} \), and \( \frac{dC}{dt} = 2\pi \frac{dr}{dt} \), we find \( \frac{dr}{dt} = \frac{12.0}{2\pi} \text{ cm/s} \approx 1.91 \text{ cm/s} \). Using \( \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \), substitute \( r = 9.07 \text{ cm} \) and \( \frac{dr}{dt} \approx 1.91 \text{ cm/s} \), we get \( \frac{dA}{dt} \approx 108.6 \text{ cm}^2/\text{s} \).

Step 5: Calculate the Induced EMF Using Faraday's Law

According to Faraday’s law, the induced electromotive force (emf) is \( |\varepsilon| = B \frac{dA}{dt} \), where \( B = 0.500 \text{ T} \). Therefore, \( |\varepsilon| = 0.500 \times 108.6 \text{ cm}^2/\text{s} \times \left( \frac{1}{100^2} \right) \approx 0.0543 \text{ V} \).

Step 6: Determine the Direction of the Induced Current

Using Lenz's Law, the induced current will flow in a direction so as to oppose the change causing it. As the loop’s area is decreasing, the induced current will flow to increase the magnetic flux. Viewed along the direction of the magnetic field, this implies a counter-clockwise current.

Key Concepts

Faraday's Law

One of the key principles of electromagnetism, Faraday's Law, describes how a changing magnetic field can induce an electromotive force (emf) in a loop of wire. Faraday discovered that the induced emf is closely related to the rate of change of the magnetic flux through the loop. In simpler terms, whenever a magnetic field surrounding a wire loop changes — whether by varying the intensity of the magnetic field or changing the loop's shape — an emf is induced in the wire. For this reason, we express Faraday's Law mathematically as:\[\varepsilon = -\frac{d\Phi_B}{dt}\]Here, \(\varepsilon\) represents the induced emf, and \(\frac{d\Phi_B}{dt}\) is the rate of change of the magnetic flux \(\Phi_B\). The negative sign signifies Lenz's Law, indicating the direction of the induced emf: it always opposes the change in magnetic flux. This principle is crucial in technologies like generators and transformers that rely upon magnetic fields and electric currents to function.

Magnetic Flux

Magnetic flux \( \Phi_B \) is the measure of the magnetic field passing through a given area, perpendicular to the field's direction. This concept is particularly important when dealing with electromagnetic induction. The magnetic flux is calculated as:\[\Phi_B = B \cdot A \cdot \cos(\theta)\]Where:- \( B \) is the magnetic field strength,- \( A \) is the area the field penetrates, and- \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface.In instances where the magnetic field is uniform and perpendicular to the surface, \( \theta = 0 \), simplifying the flux to \( \Phi_B = B \cdot A \). This formula becomes especially helpful, like in the given exercise, because any change in area or field strength results in a corresponding change in flux, thereby inducing an emf. Understanding how these elements interplay helps in interpreting Faraday's Law to predict resultant currents or voltages.

Lenz's Law

Lenz's Law is an extension of Faraday's principles, focusing on the direction of the induced current. Named after the physicist Heinrich Lenz, the law states that an induced current will always flow in a direction that opposes the change in magnetic flux that produced it. This is encapsulated in Faraday’s Law by the negative sign. For instance, if a magnetic field through a loop decreases, Lenz's Law predicts that the induced current will flow to strengthen the field back to its initial state. In our exercise, as the loop contracts, the magnetic flux through it reduces. Thus, according to Lenz's Law, the induced current would attempt to increase the flux by flowing in a manner that boosts the magnetic field across the loop. This concept is of paramount importance in shielding electronic circuits and designing magnetic braking systems, ensuring dynamic stability of electromagnetic systems.

Electromagnetic Induction

Electromagnetic induction is the process of generating an electromotive force (emf) through variations in the magnetic field around a conductor. Core to this concept is the idea that conductors moving in a varying magnetic field or magnetic fields changing around a conductor can induce voltage. This principle is exploited in many electrical devices, including:

• Electric generators, where mechanical motion is converted into electrical energy.
• Transformers, which modify voltage levels to allow efficient power transmission over distances.
• Inductive charging, utilized in wireless charging for devices.

The exercise is a classic example of electromagnetic induction: a shrinking loop in a magnetic field causes a change in the magnetic flux, which leads to an emf. Understanding how this process unfurls is fundamental to mastering electromagnetic applications and innovations.