When an electric current flows into the page, what is the direction of the magnetic field?
Answers
Explanation:
I've learnt that an Electromotive force (EMF) is induced in a loop when the magnetic flux through it changes with time. The induced EMF E is given by the Faraday's law of electromagnetic induction:
E=−dϕdt
where ϕ=∫B→.ds→ is the magnetic flux through the loop. The flux can be varied by either changing the magnetic field, or the area of the loop, or the angle between the magnetic field and the area vector of the loop.
Let us consider a metal rod of length l rotating about an end with a uniform angular velocity ω . A uniform magnetic field B→ exists in a direction perpendicular to the plane of the screen and is going into it. The free end of the rod slides along an annular metallic disk as shown in the following diagram:
As the rod is rotating about one of its ends, all points on the rod will move in a circle of radius r with a speed given by v=rω where r is the distance of the point from the axis of rotation and ω is the angular speed. Let us consider the following diagram:
The element dr is at a distance r from the axis of rotation and moves with a speed rω . Due to its speed in the magnetic field, it can be replaced by a cell of EMF dE=vBdr=rωBdr . Doing the same thing for each and every element, we can replace the entire rod with a series combination of such cells. The overall EMF of the rod between the points A and B is:
E=∫0lrωBdr=12Bωl2
On this basis, I can clearly understand why an EMF must be induced between the point at the centre and the annular disk. However, when we analyse the system using Faraday's law, it seems there must be no EMF induced as the flux remains constant (both magnetic field and the area of the loop is constant). Why is there a contradiction when we analyse the system in this way? Both methods are well established, and I don't see where I'm going wrong while analysing this using Faraday's law.