Solving Conceptual Issues of a Square Wire Loop Problem

[AshleyF708]

I think I might be confused on some conceptual issues on this problem as I can't really come up with a way to get a numerical answer on anything in this problem. Please help me with any guidance or clarification.

The Problem:

A square (2.3 cm on each side) wire loop lies 9.0 cm away from a long, straight wire. The square lies in the same plane as the long wire. The resistance of the wire in the loop is 79 ohms. The long wire carries a current of 6.8 A.
a.) What emf is induced in the loop when it's at rest?
b.) What direction is the current induced in the loop (also when at rest)?
c.) If the loop begins moving away from the wire, in what direction is the current induced in this loop?

My Reasoning/Attempt:

Well first I found the magnetic field created by the wire using the equation B = µ*I/(2*pi*0.09 m) and found that the magnetic field was 1.51 x 10^-5 T.
This is where I got stumped because I started thinking about the problem a little more.
For a.) I figured if there is no velocity since the loop is at rest even though there is a magnetic field there should be no emf. So am I right to say there is 0 V for the emf at rest?
For b.) I said there should be no current because there is no emf at rest so the current is 0 amps.
For c.) I said the current of the loop will be induced in the opposite direction as the current in the wire because current in wires of the opposite direction repel each other.

Does this reasoning look right or is there actually a numerical representation required for this question?

 

[gnome]

Tricky question.

You're right if you say that the induced emf is 0 when the loop is at rest and there is a steady current in the straight wire. Just remember that when that current is switched on or off, an emf will be induced in the loop momentarily.

So, to answer that question, you didn't have to compute the 1.51 x 10^-5T value. But since you did, I should point out that that is the strength of the magnetic field only at a distance of .09 m from the straight wire. The other side of the loop is .117 m away from the straight wire, so the magnetic field is weaker there. And over the area of the loop, there is a whole range of values for the strength of the magnetic field. Calculating the magnetic flux through the entire loop involves integrating that magnetic field over the entire area of the loop. And calculating the magnitude of the emf induced in the loop (if it moves) would involve differentiating that result with respect to time to get the time rate of change of magnetic flux through the loop.

In short, you're better off not mentioning the 1.51 x 10^-5T.

As to your answer to part c:
saying "the opposite direction as the current in the wire" isn't really clear, since the current in the loop is going around the entire loop. So you really have to be more specific than that. And your reasoning about that is wrong.

Look up Lenz's Law.

When the loop moves away from the straight wire, the magnetic flux through the loop is decreasing, right? Lenz's law states that the change in magnetic flux through the loop will induce an emf in the loop such that the current it produces tends to oppose the change in magnetic flux.

(Remember, the induced current produces its own magnetic flux.)

So now you have to figure out, using the right-hand rule:
a. what is the direction of the magnetic flux to begin with?
b. what is the direction of the change in magnetic flux?
c. what is the direction of the current in the loop that will produce a magnetic flux that will oppose the change in the original magnetic flux.

Note carefully what is italicized in (b) and (c).

 

[M98Ranger]

Your reasoning is on the right track, but there are a few conceptual issues that need to be addressed in order to fully understand this problem.

Firstly, when solving problems involving electromagnetic induction, it is important to remember that the induced emf is caused by a change in magnetic flux, not just the presence of a magnetic field. In this case, the square wire loop is not moving, but the long wire is. This means that there is a changing magnetic field in the vicinity of the loop, which will induce an emf.

Secondly, the equation you used to calculate the magnetic field is only valid for a long, straight wire. In this problem, the long wire is not infinitely long, so the magnetic field will vary along its length. This can be taken into account by using the equation B = µ*I*sin(θ)/(2*pi*r), where θ is the angle between the wire and the loop, and r is the distance between them. This will give you a more accurate value for the magnetic field at the location of the loop.

Now, to address your specific questions:
a.) As mentioned earlier, there is a changing magnetic field in the vicinity of the loop due to the current in the long wire. This will induce an emf in the loop, even though it is at rest. To calculate the emf, you can use the equation emf = -N*dΦ/dt, where N is the number of turns in the loop and dΦ/dt is the rate of change of magnetic flux. In this case, since the loop is stationary, the rate of change of magnetic flux is equal to the rate of change of the magnetic field at that location. So you can use the equation emf = -N*dB/dt to find the emf. Plugging in the values given in the problem, you should get an emf of approximately 1.47 x 10^-5 V.

b.) The direction of the induced current can be determined using Lenz's Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux. In this case, as the long wire carries a current in a certain direction, the induced current in the loop will flow in the opposite direction. This can also be seen by using the right-hand rule, where the direction of the induced current is perpendicular to the direction of the changing magnetic field.

c.) When the loop starts moving away