Question

A metal hoop is laid flat on the ground. A magnetic field that is directed upward, out of the ground, is increasing in magnitude. As you look down on the hoop from above, what is the direction of the induced current in the hoop?

Short answer

Answer: The direction of the induced current in the metal hoop is clockwise.

Step-by-step solution

Understand Lenz's Law

Lenz's law states that the direction of the induced current is such that it creates a magnetic field opposing the change in the original magnetic field. This means that if the magnitude of the magnetic field is increasing, the induced current will create a magnetic field in the opposite direction to oppose this change.

Apply the Right-Hand Rule

To determine the direction of the induced current, we can use the right-hand rule. First, extend your right hand with your fingers pointing in the direction of the increasing magnetic field (upward). Next, curl your fingers in the direction of the induced current. The direction in which your thumb naturally points will give us the direction of the magnetic field created by the induced current, which will be opposite to the upward direction.

Determine the Direction of the Induced Current

Following the right-hand rule, as the magnetic field is increasing upward out of the ground, our thumb would point downward into the ground when curling fingers in the direction of induced current. This indicates that the magnetic field created by the induced current opposes the direction of the increasing magnetic field. As you look down on the hoop from above, your fingers will curl around the hoop in a clockwise direction. Therefore, the induced current in the hoop is in a clockwise direction.

Key concepts

Induced Current

Induced current is a fundamental phenomenon in the realm of electricity and magnetism. When a conductor, such as a metal hoop, is exposed to a changing magnetic environment, an electrical current is generated within it. This creation of current is governed by Faraday's Law of electromagnetic induction, which explains that a time-varying magnetic field will induce an electromotive force (emf) in a conductor.

This can be seen in the exercise where a metal hoop experiences a change in magnetic field strength. As the field increases in magnitude, electrons within the conductor begin to move, creating an induced current. The direction of this current is such that it opposes the change that created it, a concept solidified by Lenz's Law. Understanding this opposition helps us predict the behavior of circuits in changing magnetic fields, which is crucial for designing electric generators and transformers.

Magnetic Fields

A magnetic field is an invisible force that exerts a magnetic influence on moving electric charges, magnets, and other magnetic fields. Magnetic fields are vector fields, characterized not only by their strength but also by their direction. They can be visualized as lines of force that emerge from the north pole of a magnet and curve around to enter the south pole.

In our textbook example, the hoop is subjected to an upward-directed magnetic field that is increasing in magnitude. The effect of this changing magnetic field is the generation of an induced current within the hoop. It's this very interaction between a conductor and a magnetic field that lies at the heart of many technologies, such as MRI machines and electric motors.

Right-Hand Rule

The right-hand rule is a mnemonic for understanding the orientation of vectors in electromagnetism. When determining the direction of an induced current or the force on a charged particle moving through a magnetic field, the fingers and thumb of your right hand provide a useful guide.

The rule states: position your right hand so your fingers curl in the direction of either the magnetic field for a straight conductor or the conventional current for a loop, and your thumb will then point in the direction of the resultant force or the induced current. In the exercise, your fingers represent the direction of the increasing magnetic field, and the curl of the fingers gives us the direction of the induced current's magnetic field, which is clockwise when viewed from above.

Electricity and Magnetism

Electricity and magnetism are closely related phenomena described by the unified theory of electromagnetism. Electric currents generate magnetic fields, and moving magnetic fields can induce electric currents. This interplay underlies many of the functionalities of modern day electronics and electrical systems.

In the context of our example, the increasing magnetic field generates an electric current in the hoop. This induced current itself produces its own magnetic field, which interacts with the original one according to Lenz's Law. The principles of electricity and magnetism are harnessed in devices like electric motors, where electric current produces motion through magnetic forces, and in generators, where motion is used to produce electric current through magnetic induction.