Self-inductance
In discussing the induced electromotive forces in the two coils of Figs. 17–8 or 17–9, we have considered only the case in which there was a current in one coil or the other. If there are currents in the two coils simultaneously, the magnetic flux linking either coil will be the sum of the two fluxes which would exist separately, because the law of superposition applies for magnetic fields. The emf in either coil will therefore be proportional not only to the change of the current in the other coil, but also to the change in the current of the coil itself. Thus the total emf in coil $2$ should be written2 \begin{equation} \label{Eq:II:17:31} \emf_2=\mutualInd_{21}\,\ddt{I_1}{t}+\mutualInd_{22}\,\ddt{I_2}{t}. \end{equation}  Similarly, the emf in coil $1$ will depend not only on the changing current in coil $2$, but also on the changing current in itself: \begin{equation} \label{Eq:II:17:32} \emf_1=\mutualInd_{12}\,\ddt{I_2}{t}+\mutualInd_{11}\,\ddt{I_1}{t}. \end{equation}  The coefficients $\mutualInd_{22}$ and $\mutualInd_{11}$ are always negative numbers. It is usual to write \begin{equation} \label{Eq:II:17:33} \mutualInd_{11}=-\selfInd_1,\quad \mutualInd_{22}=-\selfInd_2, \end{equation}  where $\selfInd_1$ and $\selfInd_2$ are called the self-inductances of the two coils. The self-induced emf will, of course, exist even if we have only one coil. Any coil by itself will have a self-inductance $\selfInd$. The emf will be proportional to the rate of change of the current in it. For a single coil, it is usual to adopt the convention that the emf and the current are considered positive if they are in the same direction. With this convention, we may write for the emf of a single coil \begin{equation} \label{Eq:II:17:34} \emf=-\selfInd\,\ddt{I}{t}. \end{equation}  The negative sign indicates that the emf opposes the change in current—it is often called a “back emf.” Since any coil has a self-inductance which opposes the change in current, the current in the coil has a kind of inertia. In fact, if we wish to change the current in a coil we must overcome this inertia by connecting the coil to some external voltage source such as a battery or a generator, as shown in the schematic diagram of Fig. 17–10(a). In such a circuit, the current $I$ depends on the voltage $\voltage$ according to the relation \begin{equation} \label{Eq:II:17:35} \voltage=\selfInd\,\ddt{I}{t}. \end{equation}  This equation has the same form as Newton’s law of motion for a particle in one dimension. We can therefore study it by the principle that “the same equations have the same solutions.” Thus, if we make the externally applied voltage $\voltage$ correspond to an externally applied force $F$, and the current $I$ in a coil correspond to the velocity $v$ of a particle, the inductance $\selfInd$ of the coil corresponds to the mass $m$ of the particle.3 See Fig. 17–10(b). We can make the following table of corresponding quantities.