# What does inductance mean?

I have the following notion of inductance.

When a current flows through a conductor a magnetic field is set up. Depending on the shape of the conductor we get a magnetic flux through the conductor. If the current changes, the magnetic field changes, and the flux changes. This change causes an opposite induced emf in the conductor. The emf induced divided by $\frac {di}{dt}$ gives the inductance.

As you see above my notion of conductance is only quantitative. i.e It talks about how we calculate the inductance. It doesn't explain what physically the inductance is.
For example, resistance is how much a substance can oppose the flow of current. It can be imagined as the collisions of electrons with the molecules of the conductor. It can be seen physically without any calculations. Could someone develop the same notion of inductance.

Sometimes even my understanding of inductance (as little as I know) doesn't hold up. For example:

Consider two parallel wires of unit length, each of radius a, whose centers are a distance d apart and carry equal current in opposite directions.

How will you define inductance of such a system?
If we assume the current in the first wire changes. So we calculate the flux using the magnetic field of the first wire and the area of projection of the second wire? Something just tells me this isn't all that simple.

For an inductor if a current $I$ produces a magnetic flux $\Phi$ and if the two quantities are proportional to one another the constant of proportional is the inductance $\Phi = L \, I$.

Your parallel wire example is solved here.

• I think the solved link you point to assumes no flux through the second wire? – user190199 Mar 31 '18 at 15:45

First thought, maybe you could imagine the inductance of a system as its ability to store current. The same way capacitance could be the ability to store voltage. In this sense, although its not rigorous, you could phisically picture intensity or current as the number of electrons, and voltage or potential as the energy of the electrons. So inductance could be the ability of a system to store a large number of electrons, and capacitance, the ability to store high energy electrons.

Also, maybe the equivalence between electrical and mechanical systems can help you build the intuition around these concepts.

• Capacitors store charge ($C=Q/V$); the higher the capacitance, the lower the voltage. – JEB Mar 31 '18 at 15:39
• @Jeb, capacitors don't store charge either - charge is separated in a capacitor with $Q$ charge on one plate but $-Q$ charge on the other. Capacitors store energy. – Alfred Centauri Mar 31 '18 at 18:51
• kloz, inductance isn't the ability of a system to store a large number of electrons. Inductors, like capacitors, store energy. In the case of an inductor, the inductance is a measure of the amount of magnetic energy stored for a given current through. – Alfred Centauri Mar 31 '18 at 18:53

Depending on the shape of the conductor we get a magnetic flux through the conductor.

Just to clarify what looks like a minor misconception, usually the magnetic field in the space around the conductor is more important to determining the inductance than the field inside the conductor.

As you see above my notion of conductance is only quantitative. i.e It talks about how we calculate the inductance. It doesn't explain what physically the inductance is.

Other answers have already given quantitative descriptions, so I'll stick to a qualitative one.

The way I think about it, the magnetic field stores energy. So in order to set up the magnetic field, a nonzero $IV$ product must be established over time. In order for the magnetic field to dissipate, again a nonzero $IV$ product will be produced as the energy is returned to the circuit.

The details of how much energy produces how much flux are bookkeeping details that can be summarized in the single inductance value $L$ if you're only interested in how the inductor affects the circuit around it.

Inductance is the property of a closed circuit (circuit meaning a conductor loop) to resist changes in current, specifically due to magnetic flux through the loop. To understand inductance, we first consider when inductance applies. One might begin with Maxwell's equations, and this is indeed the most practical way given that the student is familiar with them. Assuming you are not, for generality in teaching, we will only focus on quantities of interest, and how they relate to each other.

Background

The central quantities in electromagnetism are the electric field, a representation of the electric force, and the magnetic field, a vector which is perpendicular to the 'plane of action' of the magnetic force. When looking at some 2D area in space, where the electric field $$\mathbf{E}$$ and the magnetic field $$\mathbf{B}$$ exist and are constant, we can calculate the flux of the fields at (or "through") the surface by taking only the component of $$\mathbf{E}$$ or $$\mathbf{B}$$ (or any vector field) which is normal to the surface, at the surface, and multiplying this field value by the area of the surface. For example, the electric flux through a rectangle $$2\text{m}\times 2$$m square in space, when the electric field has a constant value straight through the surface of $$3$$ N/C (Newtons per Coulomb), is $$3*(2*2) = 12 = \Phi_E$$ where $$\Phi_E$$ is the electric field flux at the surface. To generalize this concept of flux beyond rectangular surfaces and constant fields, we actually take the surface integral of the vector field over the surface.

In practice, the surface is the area in the middle of a circuit loop. We find that, by Faraday's law of electromagnetic induction, if we have a closed loop of wire, and a changing magnetic field $$\mathbf{B}$$, then there is a current generated in the wire loop which is proportional to the magnetic flux (the product of the magnetic field and the loop area). Note that we are not talking about coils, or resistors, but the entire circuit area.

Inductance

Inductance is easy to define. From Faraday's law, for a circuit loop with a magnetic flux $$\Phi_M$$, there is an induced voltage $$V$$, $$V(t) = -\frac{\partial \Phi_M}{\partial t}$$ Now it's worth noting that a closed loop of wire subjected to changing magnetic flux does not have a definable potential (voltage) in the same way as a circuit with e.g. a battery. The loop is closed, but current flows - the electric field exists, but the voltage is not well-defined. You cannot select two points and calculate or a theoretical voltage drop between them.

Even still, because of the current, a voltage drop is present across e.g. resistors, and this voltage is the induced voltage in the circuit.

Consider, now, a circuit which obeys the relation, $$V(t) = L \frac{dI}{dt}$$ Where $$V(t)$$ is the induced voltage, $$I$$ is the current flowing in the loop, and $$L$$ is a constant of proportionality. We see that: $$L\frac{dI}{dt} = \frac{d \Phi_M}{d t}$$ Where the partial derivative has been replaced with a single variable derivative for simplicity, and the right hand side is positive because the EMF under consideration is the back-EMF. Therefore the proportionality constant $$L$$ is: $$L = \frac{\frac{d \Phi_M}{d t}} {\frac{dI}{dt}} = \frac{\Phi_M}{I}$$ That is, the ratio of the magnetic flux to the current in the loop. This proportionality constant exists for closed circuits, and is called the inductance of the circuit.

It's important to understand that for a circuit to have inductance, it must be closed (current flows). It's equally important to understand that components in the circuit do not have inductance, but they do contribute to the inductance of the circuit, and this contribution is called the inductance of that component.

The current in a circuit causes a magnetic field around the conductors. The magnetic field, in turn, is responsible for the magnetic flux which determines inductance of the circuit. The other contributing factor is the size of the loop - if the loop is made bigger, the flux increases, and the circuit inductance increases. Similarly, if another circuit (not connected electrically) is nearby, its magnetic field contributes to the magnetic flux in the circuit, and this increases inductance (mutual inductance).

For a circuit with current $$I$$, if the current remains the same, the magnetic field around the circuit will also remain fairly constant in amplitude. However, moving wires, using shorter or longer wires, coiling wires around each other so their magnetic fields add, all these affect the total magnetic flux through the circuit and thus the inductance. Hence, an inductor is a component which has a geometry that, with a certain amount of current flowing through it, has a known amount of magnetic flux through its area, contributing a known amount of inductance.

Finer Points and Remarks

The property of inductance is the tendency of a closed circuit to affect itself by the current flowing within itself. This follows very simple linear relationships, but involving time derivatives, which give inductance is current-impeding quality. No matter how convoluted the circuit geometry is, all that matters for the inductance is (a) the area enclosed by the circuit, (b) the current flowing through the circuit, and (c) the proximity of current carrying wires to one-another. Calculating inductance is not trivial, but observing what causes inductance can aid the design of simple inductors and the reduction of induction effects.

The reason inductance occurs is the fact that the induced voltage in a circuit is proportional to the magnetic flux changes, and the magnetic flux changes depend on the current flowing through the circuit. The logic is not circular, as it may appear, but rather is a direct result of Faraday's Law. In its common integral form, Faraday's law states that: $$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot d\mathbf{s}$$ Where $$C$$ is the path of the circuit, and $$S$$ is a surface with $$C$$ as its boundary. Defining the magnetic flux $$\Phi_M$$ as: $$\Phi_M = \int_S \mathbf{B} \cdot d\mathbf{s}$$ And the electric potential (or voltage) around a loop $$C$$ as: $$V = \int_C \mathbf{E} \cdot d\mathbf{l}$$ Faradays Law reduces to: $$V(t) = -\frac{\partial \Phi_M}{\partial t}$$ Which is the same equation as we had before, assuming $$V(t)$$ is the closed circuit potential (zero in the absence of changing magnetic fields).

The only relationship we desire for a circuit, then, is one of the form: $$V(t) = L \frac{dI}{dt}$$ That is, a linear relationship between change in current and induced voltage. As we have already shown, this is how we come to the quantity of inductance. Inductance is thus a proportionality constant between voltage and changes in current, and describes the tendency of changes in current in a circuit to change the magnetic flux through that circuit. This, through a sort of feedback process, limits the rate at which the current can change.

first things first, The case you described of two parallel wires, That can never happen because inductance depends on flux change and flux is basically no. of magnetic field lines passing perpendicularly to a given surface. and two parallel wires i mean theres is no flux involved in there. And you can understand Inductance this way. Imagine a circuit with an inductor in it since the current flowing through the inductor is constant there is no flux change and so, no induced Emf. Now we change the current steadily, this in turn causes the flux to change and there is an emf induced. And inductance is that proportionality constant that relates the two i.e "Flux and the Emf".Now inductance has got no physical meaning it is just a property of the conductor. just like capacitance.Please let me know if you are still not satisfied with the answer.

• The conductors do have volume, so we can get flux that way. – user190199 Mar 31 '18 at 15:44