Why Is Faraday's Law More Complex with Square Wires?

[teroenza]

Homework Statement

I have done a few problems with Faraday's law of induction ( closed line integral of E dot dl = -(d/dt)(magnetic flux)). Every time time I have done them, the , albeit simple, problems have used circular wires through which magnetic flux is passing through, and emf is induced into. I got to wondering why it would be more difficult to solve if the wire were a square. I know how to take line integrals and surface integrals (for the flux) of squares, but I know something would still not be right. I have 2 questions.

1. Is the induced electric field rotational, so that for a square wire, E and dl are not parallel as they are for a circular one? This would be the reason the math would get more complicated and my book stuck to circular wires.

2. Solving for E with a circular wire (or circular path of integration) can be interpreted as the electric field strength at that distance from the center of the circle. What would E represent for a square path of integration, seeing as not all the points are equidistant from the center.

I am in calc-based second semester physics, and have had mathematics up to multivariable calc.

Thank you

 

[xts]

That is a matter of symmetry. For all problems with circular wires you have translational symmetry along the wire and rotational symmetry for rotation by wire axis. So if you look at the problem in cylindrical coordinates, your solution depends only on distance from wire (r) but is independent of z and phi.

For square wire you have no such symmetry and you must solve two-dimensional equations.

 

[teroenza]

Ok, so then instead of E being a function of R alone, it would be a function of XY (or some other two coordinates). Also, E would not be parallel to dl so the dot product would be more difficult?

 

[xts]

Ok, so then instead of E being a function of R alone, it would be a function of XY (or some other two coordinates). Also, E would not be parallel to dl so the dot product would be more difficult?

More or less as you said ;)
You may always choose such loop that dl along that loop will be parallel to E, but such loop won't be a nice circle. If your integration loop is circular - then E and dl are not parallel.

PS. Your posts would be much better readable if you write your equations in $\TeX$ notation.

 

[rwisz]

for your question. Faraday's law of induction states that a changing magnetic field will induce an electric field in a conductor. This induced electric field will produce a current in the conductor, creating an electromagnetic force. The equation for Faraday's law is often written as:

∮E•dl = -dΦ/dt

where E is the electric field, dl is the path of integration, and Φ is the magnetic flux through the surface enclosed by the path of integration.

To answer your first question, the induced electric field is indeed rotational. This is because the changing magnetic field will induce a current in the conductor, creating a circular motion of charges. This circular motion of charges will then create a magnetic field, which in turn will induce another electric field. This process continues, creating a rotational electric field. This applies to both circular and square wires.

For your second question, the electric field, E, represents the strength of the induced electric field at a given point in space. For a circular wire, the electric field will be strongest at the center of the wire, where the magnetic field is changing the most. For a square wire, the electric field will also be strongest at the center, but it will have a more complex distribution due to the different distances from the center of the wire. However, the concept of the induced electric field and its relationship to the changing magnetic field remains the same for both circular and square wires.

In summary, the math may get more complicated when using a square wire due to the complex distribution of E and dl, but the underlying principles of Faraday's law remain the same. I hope this helps clarify any confusion you may have had. Keep exploring and asking questions, and good luck in your studies!