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Metals e.g. copper, have a high concentration of free electrons that can move easily and can be readily moved from their orbits. If a magnet is moved quickly through a coil of copper wire, electrons move, forming mini magnets that are called domains and electricity is generated. This effect also takes place when a coil of copper is moved through a magnet. There has to be relative motion between the coil and the magnet for an induced emf to be produced which then creates a current.

Actually it is deeply connected with the basic atomic structure. As the electrons circulate the nucleus so as a result a force is produced e.g. the gravitational filed or force of earthe is due to sun as well as its circular rotation around its own axis. So in some magnetic materials thes all atoms are alligned in such a way that they have a collective constructive force, that force is actually magnetic field.
Similarly, when we apply electricity to some material their atoms are alligned and the collective result is produced in form of strong magnetic field.
For futher understanding you can study about the Hyteresis loop, it will help you in more understanding

Moving through a magnetic field can generate electricity because of the effect of the magnet on electrons (negative charge). Using the "Left hand rule" you can see that if you have a field and a direction of movement then you create a current in the third dimension. The reason for some metals being magnetic is a little more complex and is due to the electron configuration of these metals. Hopefully this helps.

Magnets stick to materials made out of iron, nickel, or cobalt. These materials are called ferromagnetic materials. Ferromagnetic materials have a special structure. A magnet can change the direction of the magnetic forces of the atoms in these materials. The magnetic forces align and the magnet sticks to the material.

If you have a brief look at fleming's left hand rule you can see that there is a right angle relationship between movement, current and B-field (magnetic feild) when you have two it induces the third in a specific direction.

I learnt all of this in A level Physics but some courses may do this in GCSE.