Self Inductance

In this article we learn about a simple yet powerful device called inductor and learn what the self inductance is. If you convert a straight piece of wire into a coil, that piece of wire is turned into a device called inductor and it behaves quiet differently from a straight piece of wire in an electric circuit.

In Figure 1 you can see a simple electric circuit, and if you change current in that circuit, the magnetic field changes. You know that the current carrying wire has magnetic field around it. That change in magnetic field causes the change in flux through the circuit and therefore, emf is induced in the circuit in a direction to oppose the current (lenz's law). Here the emf is induced in the same circuit that has a changing current.

To demonstrate self inductance, here we consider a coil as part of a circuit and the current $i$ changes in the circuit as shown in Figure 2 (only the coil is shown). The change in magnetic field in the coil induces emf in the coil. If the coil has $N$ turns, and the average magnetic flux $\Phi_B$ for each turn in the coil is directly proportional to the current $i$, that is $\Phi_B \propto i$.

Figure 1 The change in current causes the change in magnetic flux through the circuit.
Figure 2 The emf is induced in the same coil that opposes the change in current.

Now we need a proportionality constant say $c$, so $\Phi_B = c\,i$ but it is convenient to include $N$ in the equation, so

\[N\Phi_B = Li \tag{1} \label{1}\]

where $L$ is the proportionality constant called self inductance. We'll define self-inductance later. Here $\Phi_B$ is the average magnetic flux through each coil and $N\Phi_B$ is the total magnetic flux in the coil. Rearranging the above equation gives

\[L = \frac{N\Phi_B}{i}\]

If the turns are closely wound, the inductance is proportional to $N^2$ (you know the magnetic field of a solenoid is $\mu_0nI = \mu_0NI/l$ where $l$ is the length of the solenoid). The Faraday's law tells us that the induced emf is the negative of the rate of change of magnetic flux, and therefore the emf induced in the coil is

\[\mathcal{E} = -N\frac{d\Phi_B}{dt} \tag{2} \label{2}\]

The negative sign in above equation represents the Lenz's law. The emf is always induced in a direction to oppose the increase or decrease current in the circuit. Such an opposing emf is also called back emf or counter emf. So, an inductor opposes any sudden changes in current and allows to maintain steady current in the circuit. Differentiating Equation \eqref{1} and solving with Equation \eqref{2}, we obtain,

\[\mathcal{E} = -L\frac{di}{dt} \tag{3} \label{3}\]

The self inductance is defined as the induced emf per unit rate of change of current in the circuit. Note that the induced emf is proportional to rate of change of current, no to the value of current in the circuit. If the current does not change, there is no induced emf in the inductor (coil). How to measure it? Change the current, measure the induced emf and find the self-inductance!

The SI unit of self inductance or inductance is Henry (H), $1\text{H} = 1\text{V}\cdot \text{s}/\text{A}$.

The self inductance is also called simply inductance but do not confuse this with mutual inductance. Remember strictly that if you are simply saying inductance it is self-inductance, not mutual inductance explained in the next article. The self inductance depends on the shape, size and number of turns on the coil. If a magnetic material is present, it also depends on the magnetic material.

Potential Difference in an Inductor

The change in current in an electric circuit is opposed by the induced emf in an inductor in the opposite direction. The changing current causes changing magnetic field and it causes the change in magnetic flux in the coil. According to Lenz's law the emf is induced in the direction to oppose the change in magnetic flux, and therefore, it opposes the change in magnetic field and hence current.

If the induced emf opposes the change in current, there must be the potential difference across the inductor when the current is changing. Here we need to be careful about the electric fields. There are two kinds of electric fields we are talking about right now. One kind is electrostatic electric field whose source is the source of emf in our circuit such as a battery. The electrostatic electric field $\vec E_e$ is conservative field, that is the total work done by this field over a closed path is always zero.

You saw in Faraday's law that the induced electric field is nonelectrostatic electric field $\vec E_n$ which is not conservative in nature, that is the total work done by this field over a closed path is not zero but instead that is equal to the induced emf in the coil. In other words, the nonelectrostatic electric field causes the induced emf in the coil.

We need to be careful here because there are two kinds of electric fields and whether the potential difference across inductor is real or not. To be sure we can express Faraday's law in terms of inductance as

\[\oint \vec E_n \cdot d\vec l = -L\frac{di}{dt} \]

The applied electric field $\vec E_e$ and opposing electric field $\vec E_n$ must be equal to each other. So, $\vec E_e = -\vec E_n$.

\[\oint \vec E_e \cdot d\vec l = V_{ab} = L\frac{di}{dt} \tag{4} \label{4} \]

The potential difference across the inductor $V_{ab}$ is the potential of point $a$ with respect to point $b$, that is $V_{ab} = V_a - V_b$. And we know now that there is indeed a real potential difference across the inductor even if there is non-electrostatic electric field within the inductor.

The above equation tells us that the potential difference or voltage across the inductor is proportional to the rate of change of current and inductance. The greater the inductance, greater the voltage across the inductor for a given change in current. If the current is interrupted suddenly, that leads to very high rate of change $di/dt$ (from current value to zero), that leads to very high voltage across the inductor and that kind of situation is called inductive spiking.

One important thing we can understand is that the current can not change instantaneously to zero within the inductor otherwise that leads to infinite voltage! So, the current still exists within the inductor even if the current from the power supply is interrupted and becomes zero after some time (since it can not be zero instantaneously), but within that time, the voltage gradually increases across the inductor (there is no path for current).

The voltage across the inductor can not be infinite but it is surely high enough depending on the inductance if the current is interrupted. When inductive spiking occurs, it can damage other circuit components and can be dangerous! So be careful with inductive spiking (do not try this on your own if you are not trained to work with high voltages).

The current in an inductor can not be suddenly changed. If you interrupt the current, the current still exists within the inductor and since there is no place to go, the voltage continuously builds up across the inductor leading to inductive spiking.

If you interrupt the current, the very high voltage across the inductor is created and that can destroy other circuit components and can be dangerous! If the inductance is high enough and the current is interrupted, that leads to very very high voltages (remember the voltage is proportional to both inductance and rate of change of current). So while choosing the inductor, choose it with appropriate inductance and make sure that the inductive spiking is not large enough (by choosing inductor of low inductance).

Figure 3 A flyback diode is used to solve the problem of inductive spiking.

The inductive spiking can be solved by adding a diode in parallel to the inductor allowing current to pass whenever the current is interrupted. Such a diode is called flyback diode or reverse biased diode (this diode has several names). The current gets the return path to the power supply to recharge it a little bit and it is also redirected through the inductor until the energy stored in the magnetic field is completely delivered to other sources. Beware that if you choose the wrong direction of this diode, that is if you do not connect it to appropriate terminals, the diode can be destroyed by the high voltage.

Note that the current is in the same direction even if the emf is induced to oppose it. But as the current increases or decreases, the polarity of inductor the changes. You can understand that using Equation \eqref{4}, if the current is increasing $di/dt$ is positive, and if the current is decreasing, $di/dt$ is negative, it means the the direction of current in the circuit is the same (in the sense of flow of positive charge; note that current is not the vector quantity) but the direction of induced emf of the inductor is changing. If the current does not change, the rate of change of current is zero and there is no voltage across the inductor. In this case, the inductor does not oppose the current.

An inductor opposes the change in current not the current.

Inductors help stabilize current by not letting it down to zero when the current drops to zero in ac cycle such as in the case of florescent lights. An inductor always opposes the change in current. So inductive ballast or magnetic ballast are used in series with the florescent lights. So the current is not zero and the florescent light glows continuously.

The plasma inside the florescent tube is highly nonhomic, so the increase in current leads to more ionization of gas and less resistance. The ballast helps prevent sudden changes in current and keeps the tube lit.

Inductance is highly affected by the presence of ferromagnetic materials such as iron. This affect is used to determine the number of vehicles passed in the road and indicate by traffic signals. Large inductor is placed under the road. When the vehicles pass over it, the inductance of the coil increases sufficiently due to the presence of ferromagnetic material of the vehicle (steel) which can then be used in traffic light sensors (changing the signal after a particular number of vehicles are passed). Thousands of volts? Where does this energy come from? We learn this in energy stored in magnetic field.

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