184_notes:b_flux

This is an old revision of the document!


Section 22.2 in Matter and Interactions (4th edition)

Next Page: The Curly Electric Field and Induced Current

Previous Page: What happens when you change the Magnetic Field near a loop?

In these notes, we will start thinking the right hand side of Faraday's Law (the $\frac{d\Phi_{B}}{dt}$ part) and what it means to have a changing magnetic flux. You may remember from a couple of weeks ago that we said the magnetic flux through a surface was zero. So why is it not zero here? Before, we were talking about the magnetic flux through a closed surface. (A closed surface would be a hollow shape that could hold water without spilling no matter which way you rotate it - for example a sphere, cylinder or cube.) The magnetic flux through a closed surface is always zero, even if the magnetic field or shape you chose changes. We are now talking about the magnetic flux through an open surface, which can have a non-zero value. (An open surface would be some sort of area that could not hold water without spilling - for example, a flat plane.) When the magnetic flux through an open surface changes, this can then be related to the curly electric field around the edge of that surface.

electricflux1.jpg

Before we talked about the changing magnetic flux, let's first define what we mean by magnetic flux. Just like we did before with electric flux, we will say that magnetic flux is the strength of the magnetic field over an area or rather the amount of magnetic field that goes through an area. How the magnetic field is oriented relative to the area does matter so we will again want to treat the area as vector. This is very similar to how we defined the electric flux before. Mathematically, we represent the magnetic flux as:

$$\Phi_{B}= \vec{B} \bullet \vec{A}$$

fluxsheet.jpg

where $\Phi_{B}$ is the magnetic flux (with units of $T \cdot m^2$), $\vec{B}$ is the magnetic field and $\vec{A}$ is the area vector. The area vector in this case is still the vector that has the same magnitude as the area (i.e. length times width for a rectangular area or $\pi r^2$ for a circular area) and has a direction that is perpendicular to the area, which is represented by the green arrow in the figure to the right. (This is exactly the same as with electric flux). The dot product between the $\vec{B}$ and the $\vec{A}$ then tells you about the direction of the magnetic field points relative to the area. As shown in the top figure to the left, if the magnetic field (blue arrows) and area vector (red arrow) point in the same direction, then the dot product turns into a simple multiplication. As shown in the bottom figure to the left, if the magnetic field (blue arrows) points perpendicular to the area vector (red arrows), then the dot product gives a zero. We can simplify the dot product by saying: $$\Phi_{B}= |\vec{B}| |\vec{A}| cos(\theta)$$

electricflux3.jpg

where $\theta$ is the angle between the magnetic field and the area vector. Technically, this gives you the magnitude of the flux. Flux can be positive or negative and that depends on how the area vector points relative to the magnetic field vectors.

However, in writing the magnetic flux above, we have made an assumption that the area vector points in the same direction everywhere on the surface (or rather that the area we are considering is flat rather than curved). If instead the surface is curved we would need to write the magnetic flux in terms of a very small area and then add the flux through each of those small areas together. Mathematically, we would do this with an integral: $$\Phi_B = \int \vec{B} \bullet d\vec{A}$$

As you saw in the demo video, just having a magnetic flux is not enough though - to drive a current, the magnetic flux must be changing. Mathematically, we write this change as a change in the magnetic flux over a change in time. Namely:

$$\frac{d\Phi_B}{dt}$$

If we assume that the change is happening at a constant rate, we can write this in terms of large changes (using deltas) - $ \frac{\Delta \Phi_B}{\Delta t}$. In this case, we only care about the initial and final flux relative to the initial and final time: $$ \frac{\Delta \Phi_B}{\Delta t} = \frac{\Phi_{Bf}-\Phi_{Bi}}{t_f-t_i}$$

If the change in flux is not happening at a constant rate, we then have to consider the instantaneous change in flux using the differential form: $\frac{d\Phi_B}{dt}$.

There are many ways that the magnetic flux could be changing - the size of the magnetic field could be increasing/decreasing, the size of the area could be getting bigger/smaller, the angle between the magnetic field and the area could be changing, or even some combination of these factors. The examples below go through a couple of these cases to show how you could calculate the change in the magnetic flux.

  • 184_notes/b_flux.1526404924.txt.gz
  • Last modified: 2018/05/15 17:22
  • by curdemma