Magnitude of the induced current in the loop

[chinex9a]

Homework Statement

A rectangular conducting loop of dimensions l×w and resistance R rests in the plane of a long
straight wire as shown. The closest edge of the loop is a distance a from the wire. The current in
the wire is in the upward direction and increases at a constant rate from I to 3I in time Δt.
Find the magnitude of the induced current in the loop.

Homework Equations

Maybe magnetc flux equation ABcos(theta) and magnetic field of a wire = (u*I)/(2*pi*d)

The Attempt at a Solution

Used magnetic field equation (not sure what to use for d) and don't know what to do from there.

 

[vela]

The magnetic field varies with the distance from the wire, so the magnetic field through the loop isn't a constant. That means you can't simply take some value for B and multiply it by the total area of the loop to find the total flux. You're going to have to integrate.

 

[chinex9a]

But we are trying to find the induced current. How do we come about that?

 

[vela]

That's a good question. What do you think?

 

[rwisz]

I would first clarify the units used in the given equations. The magnetic flux equation ABcos(theta) has units of magnetic field (B) multiplied by area (A) and cosine of the angle (theta). The magnetic field of a wire equation has units of magnetic permeability (u) multiplied by current (I) and divided by 2 times pi times distance (d). These equations can be used to calculate the magnetic field at a specific point due to a current-carrying wire.

To find the magnitude of the induced current in the loop, we need to use Faraday's law of induction. This law states that the induced electromotive force (EMF) is equal to the negative rate of change of magnetic flux through the loop. In this case, the magnetic flux is changing as the current in the wire increases from I to 3I in time Δt. The induced EMF will cause a current to flow in the loop, which can be calculated using Ohm's law (V=IR).

To determine the magnitude of the induced current, we need to know the rate of change of magnetic flux through the loop. This can be calculated by first finding the magnetic field at the center of the loop due to the current in the wire. This can be done using the magnetic field equation for a wire, where d is equal to the distance between the wire and the center of the loop (a + w/2). Once the magnetic field is known, we can then calculate the change in magnetic flux through the loop as the current in the wire changes from I to 3I in time Δt.

In summary, to find the magnitude of the induced current in the loop, we need to use Faraday's law of induction, Ohm's law, and the magnetic field equation for a wire. We also need to know the dimensions and resistance of the loop, the current in the wire, and the time interval Δt. With this information, we can calculate the induced EMF and use Ohm's law to determine the magnitude of the induced current.