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184_notes:b_flux [2017/11/09 03:54] – [Changing Magnetic Flux] pwirving184_notes:b_flux [2021/06/17 16:08] (current) bartonmo
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 Section 22.2 in Matter and Interactions (4th edition) Section 22.2 in Matter and Interactions (4th edition)
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 +/*[[184_notes:ind_i|Next Page: The Curly Electric Field and Induced Current]]
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 +[[184_notes:what_happens|Previous Page: What happens when you change the Magnetic Field near a loop?]]*/
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 ===== Changing Magnetic Flux ===== ===== Changing Magnetic Flux =====
-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 [[184_notes:motiv_amp_law|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. 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 (through some sort of loop), which can have a non-zero value. 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 (aka around the 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. Let's start by defining what flux is.  
 + 
 +In general, any sort of **flux is how much of something goes through an area**. For example, we could think of a child's bubble wand in terms of the air flux (from you blowing) through the circle (with the bubble solution in it). If you wanted to make bigger bubbles or make many more bubbles, you could do two things: increase the air flow or get a bubble wand with a bigger circle. Both of these actions (increasing the area and increasing the amount of air) will result in a larger "air flux" through the bubble wand. It's probably worth mentioning that we have assumed that you are holding the bubble wand so the circle is perpendicular to the air flow. If instead you rotate the wand 90 degrees, you will not get any bubbles since there is no air that is actually going through the circle part of the bubble wand. So the air flux not only depends on the amount of air and the area of circle, but also on how those two are oriented relative to each other. The idea of flux can be useful in many different contexts (i.e. fluids, electricity, air, etc.), but for any kind of flux, these are still the three conditions that matter: (1) the strength/amount, (2) the area, and (3) the orientation. 
 +  
 +Therefore **magnetic flux is the strength of the magnetic field on a surface area or rather the amount of the magnetic field that goes through an area.** For magnetic flux, we need to consider: the strength of the magnetic field, the area that the field goes through, and the orientation of the magnetic field relative to the area. These notes will introduce the mathematics behind magnetic flux, which we will use in Faraday's Law.  
 + 
 +/*  
 +This may ring some bells from what we did with electric flux last week with Gauss's Law. However, as far as we know ([[http://physicstoday.scitation.org/doi/full/10.1063/PT.3.3328|and people are looking!]]), there are no single magnetic 'charges', termed '[[https://en.wikipedia.org/wiki/Magnetic_monopole|magnetic monopoles]].' That is, everywhere we look, magnetic poles come in pairs - a north pole and a south pole. For example, if we think about enclosing a bar magnet with a Gaussian-like surface (an imaginary bubble), the magnetic flux through the whole bubble would be zero! For every magnetic field vector pointing into the bubble, there is also magnetic field vector pointing out. 
 + 
 +[{{184_notes:Week11_flux_vol.png?300|Permanent magnet enclosed in an "imaginary bubble"  }}] 
 + 
 +It turns out that this is true no matter what the source of your magnetic field is (bar magnet, moving charge, or current-carrying wire) - the integral of the magnetic field over a //closed// surface is always zero. That is, a closed volume never has net magnetic flux //__assuming that the magnetic field is constant__//. Mathematically, we would write this as: 
 + 
 +$$\oint \vec{B}\cdot d\vec{A} = 0$$  
 + 
 +Now this appears problematic. But with Gauss's law, 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. */
  
 {{youtube>RZM2uHkDyz0?large}} {{youtube>RZM2uHkDyz0?large}}
  
-==== Magnetic Flux ==== +===== Magnetic Flux ===== 
-Before we talked about the changing magnetic flux, let's first define what we mean by magnetic flux. [[184_notes:e_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 [[184_notes:e_flux#area_as_a_vector|again want to treat the area as vector]]. This is very similar to how we defined the [[184_notes:e_flux|electric flux]] before. Mathematically, we represent the magnetic flux as:+[{{  184_notes:electricflux1.jpg?250|Area Vector is perpendicular to the surface.}}] 
 + 
 +[[184_notes:e_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 [[184_notes:e_flux#area_as_a_vector|again want to treat the area as vector]]. This is very similar to how we defined the [[184_notes:e_flux|electric flux]] before. Mathematically, we represent the magnetic flux as:
  
 $$\Phi_{B}= \vec{B} \bullet \vec{A}$$ $$\Phi_{B}= \vec{B} \bullet \vec{A}$$
  
-FIXME add area vector pictures from Gauss's Law+[{{184_notes:fluxsheet.jpg?300|Flux through a horizontal plane (where dA is parallel to the magnetic field) }}]
  
-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 (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. If the magnetic field and area vector point in the same directionthen the dot product turns into a simple multiplication. If the magnetic field points perpendicular to the area vector, then the dot product gives a zero. We can simplify the dot product by saying:+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 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). You may notice that there are actually two different vectors that are perpendicular to the gray surface. We've drawn the green arrow in the $+y$ direction, but we could have also picked the vector that points in the $-y$ direction. For an open surface, it doesn't matter which of the perpendicular vectors you pick, as long as you are consistent after you pick one. 
 + 
 +The dot product between the $\vec{B}$ and the $\vec{A}$ then tells you about the direction the magnetic field points relative to the area. Mathematicallywe can simplify the dot product by saying:
 $$\Phi_{B}= |\vec{B}| |\vec{A}| cos(\theta)$$ $$\Phi_{B}= |\vec{B}| |\vec{A}| cos(\theta)$$
-where $\theta$ is the angle between the magnetic field and the area vector. Technicallythis 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.+where $\theta$ is the angle between the magnetic field and the area vector. This should also make some physical sense. As shown in the top figure to the leftif the magnetic field (shown in blue arrows) and area vector (red arrow) point in the same direction, then the dot product turns into a simple multiplication and $cos(\theta)=1$. If we look at the figure, there are lots of magnetic field arrows poking through the surface, so we'd expect a large magnetic flux. As shown in the bottom figure to the left, if the magnetic field (blue arrows) points perpendicular to the area vector (green arrow), then the dot product gives a zero since $cos(90) = 0$. If we look at the figure, we see this result as well because none of the magnetic field arrows actually go through the surface
  
-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:+[{{184_notes:electricflux3.jpg?300|Flux through a vertical plane (where dA is perpendicular to the magnetic field)  }}] 
 + 
 +(As a side notethis technically 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. This is discussed further in the [[184_notes:induced_current|page of notes on directions]].) 
 + 
 +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:
 $$\Phi_B = \int \vec{B} \bullet d\vec{A}$$ $$\Phi_B = \int \vec{B} \bullet d\vec{A}$$
  
-==== Changing Magnetic Flux ====  +===== Changing Magnetic Flux ===== 
-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:+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:
  
 $$\frac{d\Phi_B}{dt}$$ $$\frac{d\Phi_B}{dt}$$
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 [[:184_notes:examples:Week12_flux_examples|Review of Flux through a Loop]] [[:184_notes:examples:Week12_flux_examples|Review of Flux through a Loop]]
  
-Changing fluxmaybe some of the stranger changes - changing angle, etc.+[[:184_notes:examples:Week12_changing_shape|Flux Through a Changing, Rotating Shape]]
  
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