184_notes:changing_e

Section 23.1 in Matter and Interactions (4th edition)

We have spent the last two weeks talking about what happens when you have a changing magnetic field. We found that this changing magnetic field creates a curly electric field. A changing magnetic field then became another source of electric fields. You may then be wondering what happens if you have a changing electric field? We have already seen through Faraday's Law that electric and magnetic fields are related, so how do we account for a changing electric field? Perhaps unsurprisingly, a changing electric field is another source of curly magnetic fields. These notes will talk about how we amend Ampere's Law to account for a changing electric field.

From Faraday's Law, we learned that a changing magnetic field creates a curly electric field. As a similar parallel, we are now saying that a changing electric field (with time) creates a curly magnetic field. Conveniently, we already have an equation that describes a curly magnetic field: Ampere's Law. $$\int \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}$$ If you remember from a couple of weeks before, Ampere's law says that a current (the $I_{enc}$ part) will create a curly magnetic field ( the $\int \vec{B} \bullet d\vec{l}$ part). Rather than create a new equation to describe the curly magnetic field from a changing electric field, we instead just add on a term to Ampere's Law: $$\int \vec{B} \bullet d\vec{l} = \mu_0 I_{enc}+\mu_0\epsilon_0\frac{d\Phi_E}{dt}$$ where $\mu_0$ is the same constant that we have been dealing with from the last few weeks ($\mu_0 = 4\pi\cdot 10^{-7} \frac{Tm}{A}$), $\epsilon_0$ is the same constant from the first few weeks of the semster ($\epsilon_0=8.85\cdot 10^{-12}\frac{C^2}{Nm^2}$), and $\frac{d\Phi_E}{dt}$ is the change in electric flux (through the Amperian Loop).

This term that we added to Ampere's Law functions in much the same way as Faraday's law. If we can calculate the changing electric flux through a loop, then we can use that to find the magnetic field that curls around that loop. In the example below, we use a charging capacitor to illustrate how this can done; however, for the purposes of this class, we will primarily rely on this idea conceptually (rather than asking you to calculate the magnetic field from a changing electric flux).

With this final piece of the puzzle, we can actually say something really important about how electric and magnetic fields work. If we assume that there are no current-carrying wires nearby, then we have a set of two equations that say that:

A) a changing magnetic field creates an electric field $$-\int \vec{E}\bullet d\vec{l}= \frac{d\Phi_B}{dt}$$ and B) a changing electric field creates a magnetic field $$\int \vec{B}\bullet d\vec{l}= \mu_0\epsilon_0\frac{d\Phi_E}{dt}$$

Notice that there are no charges or currents anywhere in these equations. This tells us that once an electric or magnetic field is created, the charge or moving charge does not have to be present for the fields to travel through space. A changing electric field can drive a magnetic field and vice versa.

This is actually the underlying principle behind how light is created and how it travels. When a charge oscillates, it creates both a changing electric field and a changing magnetic field. Those fields then continue to oscillate through space, even at great distances away from the charge that originally created it. The oscillating electric and magnetic fields are light. This is why you may have heard light referred to as “electromagnetic waves” or “electromagnetic radiation”. (Granted visible light is only one kind of electromagnetic radiation. Depending on the frequency, electromagnetic radiation could be anything from infrared to microwaves to gamma rays.)

There are whole courses that detail how these kinds of electric and magnetic field propagate through space and interact with materials (generally listed as some version of an optics course). For our class, we will simply end with this idea that electric fields and magnetic fields are strongly related to one another and can then be used to describe light.

• Video Example: Magnetic Field from a Charging Capacitor
• 184_notes/changing_e.txt