Oersted's law
In electromagnetism, Ørsted's law, also spelled Oersted's law, is the physical law stating that an electric current creates a magnetic field. This was discovered on 21 April 1820 by Danish physicist Hans Christian Ørsted (1777–1851), when he noticed that the needle of a compass next to a wire carrying current turned so that the needle was perpendicular to the wire. Ørsted investigated and found the physical law describing the magnetic field, now known as Ørsted's law. Ørsted's discovery was the first connection found between electricity and magnetism, and the first of two laws  that link the two; the other is Faraday's law of induction. These two laws became part of the equations that govern electromagnetism, Maxwell's equations. Ørsted's rules. Ørsted found that, for a straight wire carrying a steady direct current (DC): Direction of the magnetic field. The direction of the magnetic field at a point, the direction of the arrowheads on the magnetic field lines, which is the direction that the "north pole" of the compass needle points, can be found from the current by the right-hand rule. If the right hand is wrapped around the wire so  the thumb points in the direction of the current (conventional current, flow of positive charge), the fingers will curl around the wire in the direction of the magnetic field. Vector form of the law. The above rules can be generalized to give the modern vector form of Ørsted's law The line integral of the magnetic field formula_1 around any closed curve formula_2 is proportional to the total current formula_3 passing through any surface bounded by the curve. where formula_5 = 4π×10 V·s/(A·m) is the magnetic constant, and the direction of integration around formula_2 is related to the direction of  current by the right hand rule. The law can be expressed in terms of the current density formula_7 through the surface formula_8 instead of the total current formula_3 through it where formula_8 is any surface spanning formula_2. Ørsted's law only holds for "steady" currents, which don't change with time. Therefore, it only holds for DC electric circuits, with no capacitors or inductors. It can be seen that it fails for time varying currents by considering the case of a circuit consisting of a battery charging a capacitor through a resistor. It can be verified experimentally that the current  in this circuit creates a magnetic field, yet any closed curve encircling the conductor can be spanned by a surface passing between the capacitor plates, through which no current passes, from which the equation would give zero magnetic field. Ørsted's law was modified by Maxwell to cover the case of time-varying currents by adding a new source term called displacement current, giving the Ampere–Maxwell equation.