184_notes:ind_i

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Sections 22.1-22.3 in Matter and Interactions (4th edition)

Now that we have talked about the changing magnetic flux part of Faraday's law, we should go back to the right hand side and talk about the curly electric field (the $-\int \vec{E}_{nc} \bullet d\vec{l}$ part). It is tempting to say that because we are integrating an electric field along a distance, this part of the equation would give you the electric potential. This would be true if we were integrating $\vec{E}$ and not $\vec{E}_{nc}$, but the fact that this electric field is curly means we are dealing with something else that we will call induced voltage ($V_{ind}$). These notes will discuss what we mean by $V_{ind}$, how that relates to the electric field and the corresponding current, and where the negative sign comes from.

We know from Faraday's Law that a changing magnetic field will create a curly electric field. If we put a loop of wire nearby, we can add up the little bits of length along that loop with the curly electric field, which tells us how curly that electric field is. (This is very similar to what we did with Ampere's Law.) The units of this integral ($-\int \vec{E}_{nc} \bullet d\vec{l}$) will give us the same units of electric potential (volts) - $\frac{V}{m}\cdot m= V$. However, because we have curly electric field from the changing magnetic field, this is not technically an electric potential. (When we defined the electric potential we made an assumption that all the charges were stationary or not moving, which is no longer the case.)

So instead we will define this quantity as the induced voltage. $$V_{ind}=\int \vec{E}_{nc} \bullet d\vec{l}$$ where the induced voltage is adding up the curly electric field around a loop.

Perhaps more intuitively, we can also define the $\int \vec{E}_{c} \bullet d\vec{l}$ in terms of the current induced in the wire since the curly electric field in the loop will push the charges and create a current. Using the familiar Ohm's law, we can say that $$V_{ind}=I_{ind}R=\int \vec{E}_{nc} \bullet d\vec{l}$$ Where $I_{ind}$ is the current that is created in the loop of wire and R is the resistance of that wire. When we put this together with the rest of Faraday's law, this means that we can write this equation in three different ways: \begin{align*} -\int \vec{E}_{nc} \bullet d\vec{l} &= \frac{d \Phi_B}{dt} &&&& (1) \\ -V_{ind} &= \frac{d \Phi_B}{dt} &&&& (2) \\ -I_{ind}R &= \frac{d \Phi_B}{dt} &&&& (3) \end{align*} Depending on what you want to focus on or calculate you may pick one of these equations, but they are saying the exact same thing.

When we're talking about Faraday's Law, the negative sign plays an important role. It must be there to satisfy momentum and energy conservation, which we'll talk about in the next few paragraphs. If we think about the third equation above, the negative sign says that the induced current generates a magnetic field that will be in the direction that opposes the change in magnetic flux. The curly electric field that the changing magnetic field generates points in the same direction as this current. This should make some intuitive sense because the conventional current always points in the same direction as the electric field that is driving it. However, we generally talk about the direction of the induced current (rather than the electric field or induced voltage) since it make more intuitive sense and is easy to measure.

As an example of how to figure out which direction the induced current flows, let's say we have a bar that is sliding down a pair of connected conductive rails (so current is free to flow through the loop created by the bar and rails), which is sitting in a magnetic field that points into the page (shown in the top figure to the right). Initially there would be some magnetic flux through the loop (directed into the page following the magnetic field). At a later time (shown in the second picture to the right), after the bar has moved down the rails, there would be a larger magnetic flux through the loop (still directed into the page following the B-field) because the area of the loop will have increased. Since the magnetic flux increased, we know that there should be an induced current in the loop - but what direction should it flow around the loop? We can use another right hand rule to figure it out.

For this right hand rule, you want to first point your right thumb in the direction of the change in magnetic flux. For our example, the change in magnetic flux would point into the page. (Because $\Delta \Phi_{B}= \Phi_{Bf}-\Phi_{Bi}$, we would have a large flux into the page as the final minus a small flux into the page as the initial, which leaves the change in flux as into the page.) Because the induced current would point opposite to the change, you would flip your thumb in the opposite direction, so pointing out of the page. Finally the direction that your fingers would curl in would be the direction of the induced current. So in our example, the current would flow counter-clockwise around the loop (shown in the figure to the left).

As we said before, the fact that the induced current will always generate a magnetic field to oppose the change in flux is an important result and ties back to energy and momentum conservation (it even is sometimes referred to as it's own law: Lenz's law). We can see this by thinking about what actually happens to the bar as it is moving through the magnetic field. Now that there is a induced current flowing through the bar, it would also feel a magnetic force from the magnetic field. Using our right hand rule again, this time with the $I_{ind}$ (point your fingers in the direction of the induced current) and the B-field (curl your fingers toward the B-field), we would get a force (from your thumb) that would point opposite to the velocity of the bar (shown in the figure to the right). So the magnetic force on the bar with the induced current would actually work to slow the bar down. If this negative sign were not present, the induced current would flow clockwise instead of counter-clockwise, thus making the magnetic force on the bar point in the same direction as the velocity. This would act to speed up the bar and give us a way to get more energy than we started with, which would violate the conservation of energy.

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