What is the difference between a bar magnet and an electromagnet? How can a magnetic field be defined by 2 different phenomena?
A bar magnet produces magnetic field due to its pole strength (+m & -m) but a current carrying element produces magnetic field due to moving charges.
Are these 2 the same thing or different?
It will be helpful if you try to explain it without going into special relativity, or higher level calculus.
 A: Electricity and Magnetism are the two sides of the same coin. Magnetism is produced only due to the effect of moving charges, every charged body in motion acts like a magnet. Therefore, in the atoms of a magnet, the electrons moving around their nucleus are actually magnet! Now one would ask if it was so, then all the elements around us should act like magnet because all of them have electrons. But, the answer is - NO! It is because even if an atom as a whole is magnetic but they all are aligned in such a manner that their net magnetic effect is cancelled out and the element is non magnetic.
In certain elements, the atoms are aligned in such a manner that the magnetic effect do not cancel out. Such elements are known as - "Ferromagnetic Elements" Therefore the magnetic field of a bar magnet is not because of its pole strength but actually because of its perfect alignment of magnetic atoms and that the atoms are magnetic because of the negatively charged moving electrons.
A: Aditya, they are not the same. There is no such thing as a bar magnet whose magnetic moment is due to electric currents alone. A bar magnet is composed of a ferromagnetic material such as iron and its magnetism is mainly due to electron spins (spin is a fundamental quantum property and there is no evidence that it involves any actual motion). The orbit of each electron also contributes to the magnetic moment of an atom or ion, but in the transition metals (which include iron) it is quenched and is small enough to ignore. There can be "eddy" currents in magnetic metals and alloys, but they contribute little to the magnetism. Thus, a bar magnet is quite different from an electromagnet, and its magnetism is due to electron spins, not electric currents.
Spins of electrons in neighboring atoms are aligned by the exchange interaction, which is a consequence of the Pauli exclusion principle. This ordering can give rise to ferromagnetism as long as the temperature is not too high.
For more information, see Chapter 15 of Kittel, "Introduction to Solid State Physics", 6th ed., John Wiley & Sons 1986; Chikazumi, "Physics of Ferromagnetism", 2nd ed., Oxford, 1997; and Spaldin, "Magnetic Materials: Fundamentals and Applications", 2nd ed., Cambridge, 2011.
