electromagnetismhomework-and-exercisesmagnetic fields
The above Question asks about the Magnetic Flux through the Plane x<0 by the Current Carrying Square Loop.
Now, I came across a trick where it goes as:
The Plane can be replaced by a wire carrying current "I" along y-axis, and the Flux of Magnetic Field of that Wire through the given Loop will be the same as our required one.
So, my question is that is it valid?…
And if Yes, then what's the logic behind that?
Moving charge always produces a magnetic field. If you have a non-zero current then you have non-zero moving charge and a magnetic field will be produced.
You can achieve essentially no magnetic field though by using two wires right next to each other each carrying current in the opposite directions. As long as the wires are very close and the amount of current they carry is very close the magnetic fields they produce will nearly cancel. This is why a clamp meter can't measure current around two conductors carrying current in opposite directions.
I'm not sure I understand how the loop is set but your case must be similar to the left and right side of the loop below.Since the current is flowing in a loop, on the left side it will flow the opposite direction of the way it does on the other side.Using the right-hand rule you will now see that the forces are equal and opposite,as shown in the picture, explaining the minus in your problem.
So, watch out which direction the current is going, always use the R.H.R. and carefully you will have no problem.
Best Answer
Imagine the shaded area is a very big square loop of wire. The mutual inductance $M$ of the two loops is the magnetic flux $\Phi$ through one loop due to a current $I$ in the other, divided by $I$. $M$ can be calculated from the flux through the bigger loop due to a current in the smaller loop (this is what is being asked), or from the flux through the smaller loop due to a current in the bigger loop: the result will be the same. It is easier to evaluate the latter, however. As a corollary, the flux through the bigger loop due to a current $I$ in the smaller one is the same as the flux through the smaller loop due to a current $I$ in the bigger loop.
When a current $I$ flows in the bigger square loop, the three sides of this square are too far away from the small loop to contribute to the flux through it, and only the closest section along the y-axis matters. Therefore, the flux can be calculated by considering only the magnetic field due to an infinite wire along the y-axis.