Calculating Induced Current in a Rectangular Loop

[shannon]

Homework Statement

A WxH rectangular loop of wire, with resistance R, lies on a table a distance s from a separate long straight wire carrying a current I.
If the loop is pulled to the right, parallel to the wire, with the speed v, then what is the magnitude of the current induced on the loop?

Homework Equations

The Attempt at a Solution

I found that (a distance r away)
dφ=B•dA=(µₒI/2π)Wdr
now, to find φ
φ=(µₒI/2π)∫dr/r
I'm not sure about these limits...I was thinking from W->W+c (where c is just a constant)
But I'm not sure...
From here I was going to use the equations:
ε= -dφ/dt
I=ǀεǀ/R

 

[alphysicist]

Hi shannon,

Homework Statement

A WxH rectangular loop of wire, with resistance R, lies on a table a distance s from a separate long straight wire carrying a current I.
If the loop is pulled to the right, parallel to the wire, with the speed v, then what is the magnitude of the current induced on the loop?

Homework Equations

The Attempt at a Solution

I found that (a distance r away)
dφ=B•dA=(µₒI/2π)Wdr
now, to find φ
φ=(µₒI/2π)∫dr/r
I'm not sure about these limits...I was thinking from W->W+c (where c is just a constant)
But I'm not sure...
From here I was going to use the equations:
ε= -dφ/dt
I=ǀεǀ/R

Was there more information given? I think it matters which side of the rectangle is parallel to the long wire.

Also, I think it would be better to consider this as a motional emf problem rather than directly calcuating the loop flux. Does that help?

 

[thomate1]

to calculate the induced current on the loop.

I would first clarify the variables and assumptions in this problem. The WxH rectangular loop of wire is assumed to have a constant resistance R and be lying flat on a table a distance s away from a long straight wire carrying a current I. The loop is then pulled parallel to the wire with a speed v. We are asked to calculate the magnitude of the induced current on the loop.

To solve this problem, we can use Faraday's law of induction which states that the induced electromotive force (emf) in a closed loop is equal to the negative rate of change of the magnetic flux through the loop. In this case, the magnetic flux through the loop is changing as it moves parallel to the wire, so an emf will be induced.

Using the equation for magnetic flux, φ=B•A, where B is the magnetic field and A is the area of the loop, we can find the induced emf by taking the derivative with respect to time. This gives us ε=-dφ/dt. Since we are interested in the current induced in the loop, we can use Ohm's law, I=ε/R, where R is the resistance of the loop.

To find the magnetic field, we can use the Biot-Savart law which states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire. In this case, the magnetic field at a point on the loop due to the current in the straight wire can be calculated as B=(µₒI/2πr), where µₒ is the permeability of free space and r is the distance from the wire to the point on the loop.

To calculate the area of the loop, we can use the dimensions given in the problem, W and H. However, since the loop is moving parallel to the wire, the area of the loop will change as it moves. We can use the equation for the area of a rectangle, A=WxH, and take the derivative with respect to time to find the rate of change of the area, dA/dt=Wdv/dt. This gives us A=WHv. Substituting this into the equation for magnetic flux, we get φ=B•A=(µₒI/2π)(WHv). Taking the derivative with respect to time, we