The first coil has N1 turns and carries a current I1 which gives rise to a magnetic field B1 G . Since the two coils are close to each other, some of the magnetic field lines through coil 1 will also pass through coil 2. Let Φ21 denote the magnetic flux through one turn of coil 2 due to I1. Now, by varying I1 with time, there will be an induced emf associated with the changing magnetic flux in the second coil: 21 21 2 1 2 coil 2 d d N dt dt ε d Φ = − = − ⋅ ∫∫ B A G G (11.1.1) The time rate of change of magnetic flux Φ21 in coil 2 is proportional to the time rate of change of the current in coil 1: 21 1 2 2 d N M dt dt 1 Φ dI = (11.1.2) where the proportionality constant M21 is called the mutual inductance. It can also be written as 2 21 21 1 N M I Φ = (11.1.3) The SI unit for inductance is the henryLet’s consider the process involved in creating magnetic energy. Figure 11.3.1 shows the process by which an external agent(s) creates magnetic energy. Suppose we have five rings that carry a number of free positive charges that are not moving. Since there is no current, there is no magnetic field. Now suppose a set of external agents come along (one for each charge) and simultaneously spin up the charges counterclockwise as seen from above, at the same time and at the same rate, in a manner that has been pre-arranged. Once the charges on the rings start to accelerate, there is a magnetic field in the space between the rings, mostly parallel to their common axis, which is stronger inside the rings than outside. This is the solenoid configuration. Figure 11.3.1 Creating and destroying magnetic field energy. As the magnetic flux through the rings grows, Faraday’s law of induction tells us that there is an electric field induced by the time-changing magnetic field that is circulating clockwise as seen from above. The force on the charges due to this electric field is thus opposite the direction the external agents are trying to spin the rings up (counterclockwise), and thus the agents have to do additional work to spin up the charges because of their charge. This is the source of the energy that is appearing in the magnetic field between the rings — the work done by the agents against the “back emf.” Over the course of the “create” animation associated with Figure 11.3.1, the agents moving the charges to a higher speed against the induced electric field are continually doing work. The electromagnetic energy that they are creating at the place where they are doing work (the path along which the charges move) flows both inward and outward. The direction of the flow of this energy is shown by the animated texture patterns in Figure 11.3.1. This is the electromagnetic energy flow that increases the strength of the magnetic field in the space between the rings as each positive charge is accelerated to a higher and higher speed. When the external agents have gotten up the charges to a predetermined speed, they stop the acceleration. The charges then move at a constant speed, with a constant field inside the solenoid, and zero “induced” electric field, in accordance with Faraday’s law of induction. We also have an animation of the “destroy” process linked to Figure 11.3.1.Over the course of the “destroy” animation associated with Figure 11.3.1, the strength of the magnetic field decreases, and this energy flows from the field back to the path along which the charges move, and is now being provided to the agents trying to spin down the moving charges. The energy provided to those agents as they destroy the magnetic field is exactly the amount of energy that they put into creating the magnetic field in the first place, neglecting radiative losses (such losses are small if we move the charges at speeds small compared to the speed of light). This is a totally reversible process if we neglect such losses. That is, the amount of energy the agents put into creating the magnetic field is exactly returned to the agents as the field is destroyed. There is one final point to be made. Whenever electromagnetic energy is being created, an electric charge is moving (or being moved) against an electric field ( ). Whenever electromagnetic energy is being destroyed, an electric charge is moving (or being moved) along an electric field .