Considering an Inductor in a DC circuit: when the switch is first closed there is a change in current through the inductor, which induces an emf in the opposite direction (Lenz's Law). My question is why does that emf cause a decrease in the rate of change of current through the inductor? (which would then cause a reduced emf and then an increased rate of change of current?)
My question is why does that emf cause a decrease in the rate of change of current through the inductor
The emf doesn't 'cause' a change in the rate of change of inductor current but it is, however, consistent with the rate of change in inductor current.
The fact is that the rate of change of current, the associated rate of change of magnetic field, and the associated non-conservative electric field must satisfy Maxwell's equations (the governing equations of classical electromagnetism) at each instant of time.
For the case that there is a constant, non-zero voltage across the inductor, Maxwell's equations are satisfied when
- the inductor current changes at a constant, non-zero rate
- the magnetic field changes at a constant rate, non-zero rate
- the non-conservative electric field is constant and non-zero (which implies a constant, non-zero emf).
So it isn't true that the emf causes a change in rate of change of inductor current since, as pointed out above, there is a perfectly consistent solution in which the emf is non-zero and the rate of change of current is constant.
I will describe how to make something that acts the way the inductor acts so then it should be clear.
Imagine you didn't have the inductor and you replaced it with a wire. Then current would flow through the wire.
Now imagine that there are tons of little batteries there not hooked up. Some pointing one way some pointing the other way. Now a battery only has so much energy stored in it, if you run a current through it too long the battery fails and dies.
OK so you replaced the inductor with a wire and a bunch of batteries and we will say that the amount of energy stored in your batteries is proportional to the square of the current going through that section of wire.
Imagine you hired someone to make that happen. When the current increases that person would have to hook up a bunch of batteries facing the direction where the current flows the wrong way (so the batteries can get energized) and would have to hook up enough batteries so they get charged at the rate that keeps up with how much energy they are supposed to have stored for the increasing current.
And what if the current decreased then the energy stored would have to go down, so the person you hired would have to hook up the right number of batteries in the direction where they lose energy.
That means there are a bunch of tiny batteries pointing one way or another way and the number of them that is hooked up is related to how quickly the current is changing. And they are wired up to gain or lose energy and they gain energy when the current increases and they lose energy when the current is decreasing.
So we know how such a circuit element works. It has a current and it has a bunch of batteries that have a stored energy proportional to the square of the current and that the batteries may or may not be hooked up and different amounts could be hooked up and they could be wired to energize themselves or to energize the charges flowing through them. But you have more of them hooked up the more the current changes. Enough to let the energy always be proportional to the square of the current.
That's what an inductor is. The energy is stored in the magnetic field (magnetic fields have energy and twice the field is four times the energy). And so if the energy of the magnetic field changes that energy must go somewhere or come from somewhere. And it goes to the charges or comes from the charges.
We can represent that exchange of energy by an emf (think of it as power per current). And that it because it literally does do work on electric charges at just that right rate to balance the energy.
So how does the emf work. It works exactly like a bunch of batteries. So you can imagine the inductor is a battery whose voltage is related to the rate of change of the current. Because a changing current means a changing magnetic field hence changing amount of energy in the magnetic field and hence the energy must come from somewhere and the energy comes from the charges (or go somewhere).
You can literally imagine that there is a battery whose voltage depends on the rate of change of the current. And there is a good reason why. When you run current backwards through a battery you energize the battery. And you are energizing the inductor. The larger magnetic field in the inductor literally has more energy.
So circulating electric fields are associated with changing magnetic fields and hence with channeling energy into magnetic fields.
So the battery is energizing charges which flow to the inductor and it is there at the inductor that energy flows into the inductor (into its magnetic field) when the current is increasing (and energy flows from the magnetic field in the inductor to the charges when the current is decreasing).
So magnetic fields have energy and it takes energy to increase them and you get energy when you decrease them. Since the magnetic field is associated with the current, the changing current is associated with a changing magnetic field so energy for that comes from somewhere it comes from the charges.
And how do we quantify energy going to or coming from the charges? With an emf. And the direction is such as to balance the energy. Decrease the energy of the charges (like energizing a battery).
And note that when I say energize a battery I mean like recharging a rechargeable battery I'm not talking about changing the strength. You can imagine that there are tons of batteries each of a fixed voltage and each facing a certain way and they get wired in when current changes, in a way to suck energy if the current increases and a way to give back energy if the current decreases and if the change is more rapid you connect more of them. So they act just like batteries and we put in enough of them to correctly account for the energy that exchanges between the charges and the magnetic field.
In fact you can think of all electrodynamics as the fields having energy and momentum and where there are no charges or currents the momentum and energy flows around as the fields change and even as the fields sit there (and yes energy and momentum can flow even when the fields are static just like electric current can flow even when the electric charge density isn't changing). So normally the energy and momentum of the fields just flows around. But when there is current and there is charge then the fields can lose or gain energy and they can lose or gain momentum and the charges balance it out.
So if your circuit isn't moving an emf is just a power per current so for your given current you can find out how much emf you need to account for the energy exchanging with the magnetic field. A similar thing happens with a capacitor. When it is charged there is an electric field and that has energy. The voltage across the capacitor is exactly that needed to balance the energy taken from the charges to give to the field.
So your real battery energizes the charges and they go around the circuit and as they approach the capacitor the charge already there takes the energy of the charges and stores it in an increased electric field. You can think of it as when a charge feels a force from an electric field where does the energy go to or come from? It goes into the energy of the fields. So if field energy flows into a region then either an equal amount better flow out or else he field energy there must change or else energy must exchange right then and there with a charge.
OK. Let's be honest about why you didn't already know this, because it's not your fault. In electrostatics you had charges exert forces instantaneously on each other and you had potential energy. But all of that is incorrect. It only approximately holds when things don't move. The more complicated reality is that there is energy and momentum in the charges and also in the fields spread throughout space. And it flows around and it can be exchanged between the charges and currents and the fields.
Are the fields changing their energy because the charges are changing their energy? Or the other way around? Neither is entirely accurate. They must happen together. Energy conservation says they happen together. And then we have to make specific laws about how the interaction between charges, currents, and fields. That interaction dictates how they both must change. It isn't one causing the other.
In our example where you paid someone to connect the batteries the changing current it seemed like one caused the other. Imagine you paid them but they didn't do anything because they knew the magnetic field would store the energy for them.
The reality is the charges move in a way to vain or lose energy and the fields are such as to interexchange energy without themselves to have the fields lose energy or gain energy right where those charges are and of the exact right rate to balance out energy conservation But each just interacts the way it does.
And momentum does the same thing except current can be involved in momentum exchange without exchanging energy.
Lenz's Law says that in any conductor the current flows in first cycle it was take more current due to more magnetizing the coil that is before starting the first half cycle , at the same time the flux is again cross over the conductor emf is induced in it (same coil), so induced emf is opposite direction to oppose the current up to half cycle.
After 90 degrees , the same flux is in induced in same direction so that time current is twice up to 180 degrees.