In 1820 Hans Oersted made a compass needle move by placing a current-carrying wire nearby. This simple act clearly demonstrated that electricity and magnetism are linked. In 1831, Michael Faraday added to our understanding of this electromagnetic link when he produced an electric current from a magnetic field. He connected one coil of wire to a battery and a second coil of wire to nothing but a galvanometer to detect any current. His expectation was that the current in the first coil would produce a sufficient magnetic field to produce a current in the second coil. When he applied a steady current to the first coil, it did not induce a current in the second coil. But when he closed the switch in the first circuit, the magnetic field strength changed, which created a current in the second coil. From this experiment Faraday concluded that though a steady magnetic field produces no current, a changing magnetic field can produce an electric current. This phenomenon is known as electromagnetic induction. Faraday conducted further experiments on this phenomenon. Let's reproduce one of its experiments by using a magnet, a coil of wire, and a galvanometer. Note that when we move the magnet toward the coil, the magnetic field strength increases, and this increase in the magnetic field induces a current in the loop. If the magnet is moved away from the coil, the magnetic field decreases in strength, and the current in the coil reverses and moves in the opposite direction. No current is induced when the magnet does not move relative to the coil. In such a case, the galvanometer reads zero. We can conclude from this experiment that motion is required to produce a current. Based on this two experiments Faraday showed that, simply by changing the magnetic flux, current can be induced in a coil that is not physically connected to anything. The magnetic flux through the area bounded by a conducting loop is proportional to the number of magnetic field lines passing through the loop. Since the number of field lines per unit area is proportional to the strength of the field, the magnetic flux is equal to the product of the magnitude of the magnetic field and the surface area threaded by the field. Consider the simple case of a planar loop perpendicular to a magnetic field that is uniform in both magnitude and direction. Using the Greek letter Phi sub B to represent the magnetic flux we obtain: Phi sub B equals B A where B is the strength of the field and A the area bounded by the loop. Now let's consider the case in which the loop is not perpendicular to the magnetic field. It is obvious that fewer field lines are threading the loop than in the previous case. The magnetic flux through the loop decreases as we rotate the loop. Ultimately- when we have rotated the loop until it lies parallel to the direction of the field - the flux is zero. Notice that the number of field lines threading the loop is equal to the number of field lines that go through the shaded area A prime, which is perpendicular to B. Thus the magnetic flux through the loop can be expressed as Phi sub B equals B A prime. If we use theta to represent the angle between the normal to the loop, n, and the direction of the magnetic field, we can represent the relation between the area A prime and A by the equation A prime equals A cosine theta. Therefore the magnetic flux through a loop of area A, called phi sub B, is given by the product of the magnetic field strength times the loop's area times the cosine of theta. The magnetic field strength is expressed in Teslas, and the Area, in square meters. Therefore the unit of the magnetic flux are Teslas square meters. This unit is known as the Weber. From this result, we find, again, that when theta is zero degrees, cosine of theta is one, and the magnetic flux is at its maximum. When theta is 90 degrees, cosine of theta is zero, and the magnetic flux is zero. If the flux through N loops of wire changes by an amount d Phi B during a time d t, the induced e m f during this time is written epsilon equals minus the number of loops N times d Phi B divided by d t. This equation is known as another form of Faraday's law of induction. If the flux through N loops of wire changes by an amount d Phi B during a time d t, the induced e m f during this time is written epsilon equals minus the number of loops N times d Phi B divided by d t. This equation is known as another form of Faraday's law of induction. If the secondary coil includes more than one loop, then the change in flux is multiplied by N, the number of loops. The e m f, or epsilon, is a voltage and has units of Volts. Suppose that the north end of a magnet is moved toward a 100-loop solenoid with a radius of 25 centimeters, causing the magnetic field to increase from zero to 2.3 Teslas in 0.75 seconds. What is the induced e m f in the solenoid during this change? A conducting loop consisting of half a circle of radius 0.40 meters lies in a uniform magnetic field B that is directed out of the screen; the magnetic field magnitude is given by B equals 8 t squared plus 4 t plus 6. A battery of 2.0 volts is connected to the loop. The resistance of the loop is 2 Ohms. What is the magnitude and direction of the e m f induced along the loop by B at t equals10 seconds? What if you had a 100-turn, square solenoid in a magnetic field and then pulled it quickly out of the field? It would take energy to pull the solenoid out of the field, and this same amount of energy would be found in the newly created current in the loop. You can think of this as the solenoid's resistance to being pulled out of the field. Let's review some formulas that may prove useful later. Power is current times energy and is expressed in Watts or Joules per second. Energy is current times resistance. Suppose this solenoid is 12 centimeters on a side, has a resistance of 75 Ohms, and is pulled out of the 5.0 Tesla field in 0.25 seconds. What is the induced e m f? How would a graph of e m f versus distance look if you moved the loop from left to right through the magnetic field? Notice that voltage is produced only when the magnetic flux is changing. In section b, the loop is entering the field and the e m f is positive. In section d, the loop is exiting the field and the e m f is negative. The e m f is zero when the loop is entirely out of or in the field.

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