184_notes:b_flux

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184_notes:b_flux [2021/06/17 15:26] – [Changing Magnetic Flux] bartonmo184_notes:b_flux [2021/06/17 16:08] (current) bartonmo
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 {{youtube>RZM2uHkDyz0?large}} {{youtube>RZM2uHkDyz0?large}}
  
-==== Magnetic Flux ====+===== Magnetic Flux =====
 [{{  184_notes:electricflux1.jpg?250|Area Vector is perpendicular to the surface.}}] [{{  184_notes:electricflux1.jpg?250|Area Vector is perpendicular to the surface.}}]
  
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 $$\Phi_{B}= \vec{B} \bullet \vec{A}$$ $$\Phi_{B}= \vec{B} \bullet \vec{A}$$
  
-[{{184_notes:fluxsheet.jpg?300|Flux through a horizontal plane (dA is parallel to the magnetic field) }}]+[{{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 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. 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.
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 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 left, if 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.  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 left, if 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. 
  
-[{{184_notes:electricflux3.jpg?300|Flux through a vertical plane (dA is perpendicular to the magnetic field)  }}]+[{{184_notes:electricflux3.jpg?300|Flux through a vertical plane (where dA is perpendicular to the magnetic field)  }}]
  
 (As a side note, this 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]].) (As a side note, this 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]].)
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 $$\Phi_B = \int \vec{B} \bullet d\vec{A}$$ $$\Phi_B = \int \vec{B} \bullet d\vec{A}$$
  
-==== 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:
  
 $$\frac{d\Phi_B}{dt}$$ $$\frac{d\Phi_B}{dt}$$
  • 184_notes/b_flux.1623943568.txt.gz
  • Last modified: 2021/06/17 15:26
  • by bartonmo