Shrinking Loop. 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. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Verified step by step guidance

1. Step 1: Understand that the loop's circumference is decreasing, which means the area of the loop is also decreasing.
2. Step 2: Use Faraday's Law of Induction, which states that an electromotive force (emf) is induced in a loop when there is a change in magnetic flux through the loop.
3. Step 3: Calculate the rate of change of the area of the loop. The circumference C is given by C = 2πr, and the area A is given by A = πr^2. Since the circumference is decreasing at a rate of 12.0 cm/s, find the rate of change of the radius and then the area.
4. Step 4: Determine the direction of the induced current using Lenz's Law, which states that the direction of the induced current will be such that it opposes the change in magnetic flux. Since the magnetic field is directed into the page (as indicated by the 'x' symbols), and the area is decreasing, the induced current will create a magnetic field that opposes this decrease.
5. Step 5: Apply the right-hand rule to find the direction of the induced current. Point your thumb in the direction of the magnetic field (into the page), and curl your fingers in the direction that would create a magnetic field opposing the decrease in flux. This will give you the direction of the induced current as viewed along the direction of the magnetic field.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The induced EMF is proportional to the rate of change of the magnetic flux. In this scenario, as the circumference of the loop decreases, the magnetic flux through the loop changes, leading to the generation of an induced current.

Lenz's Law

Lenz's Law states that the direction of the induced current will be such that it opposes the change in magnetic flux that produced it. This means that if the loop is shrinking and the magnetic flux is decreasing, the induced current will flow in a direction that attempts to maintain the original magnetic flux through the loop.

Magnetic Field Orientation

The orientation of the magnetic field is crucial in determining the direction of the induced current. In this case, the magnetic field is perpendicular to the plane of the loop. By applying the right-hand rule, one can determine the direction of the induced current based on the orientation of the magnetic field and the change in area of the loop.