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 induced current in the hoop is clockwise.

Step-by-step solution

Understand Faraday's Law

Faraday's Law of electromagnetic induction states that the electromotive force (EMF) induced in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it is expressed as: \(\text{EMF} = - \frac{d \Phi_B}{dt}\) Where \(\Phi_B\) is the magnetic flux and \(t\) is time. The induced current in the hoop will create a magnetic field that opposes the change in the external magnetic field, as stated by Lenz's law.

Determine the direction of the change in magnetic flux

In this case, the magnetic field is increasing in magnitude and is directed upward, out of the ground. This means that the change in magnetic flux is also upward.

Apply the right-hand rule

To determine the direction of the induced current, we can use the right-hand rule. The right-hand rule states that if we point our thumb in the direction of the magnetic field and curl our fingers, the direction of our fingers represents the direction of the induced current. Since the change in magnetic flux is upward, point the thumb of your right hand upward. Curl your fingers around the hoop, and the direction in which your fingers curl is the direction of the induced current.

Determine the direction of the induced current

From Step 3, when pointing your right thumb upward, your fingers will curl around the hoop in a clockwise direction (when looking down at the hoop from above). Therefore, the induced current in the hoop is clockwise.