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In the figure below, from a textbook explaining Eddy Currents, a pulse of current is sent and generates a magnetic field $ B_0 $. The magnetic field produces a change in flux in the metal box that generates an electromotive force, driving a current and therefore generating another magnetic field $ B' $
However, I don't understand how this induced magnetic field generates another current $ I' $ in the opposite direction in the receiver coil of the metal detector.
A pulse of current would produce a magnetic field which would increase and then decrease. The eddy currents would go first in one direction and then the other. Each of these would induce a voltage in the coil first in one direction and then the other. To analyze directions you will have to decide which of these parts of the cycle you wish to deal with. I get the impression from my detector which outputs a variable tone, that an AC current in the coil resonates with a capacitor. A piece of metal within the field of the coil changes the inductance of the coil and the frequency of the tone. (Actually, the tone may be a beat frequency that results from adding the voltage of a tunable circuit to that from the coil circuit.)
Changing magnetic field induces curling electric field and the induced current is in the direction of electric field. We use right hand rule where thumb points in the direction of changing magnetic field and curl gives due direction of current. $$\vec{B_0(t)}=-\vec{B'(t)}$$ Both magnetic field produces currents curling in opposite direction.
