184_notes:examples:week12_changing_shape

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184_notes:examples:week12_changing_shape [2018/04/11 19:38] – [Solution] pwirving184_notes:examples:week12_changing_shape [2018/08/09 18:08] (current) – curdemma
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 +[[184_notes:b_flux|Return to Changing Magnetic Flux notes]]
 +
 ===== Flux Through a Changing, Rotating Shape ===== ===== Flux Through a Changing, Rotating Shape =====
 Suppose you have a magnetic field directed in the $-\hat{z}$-direction, into the page. There is a flexible, circular loop situated on the page, in the $xy$-plane. You stretch it out in the $\pm x$-direction like a rubber band to change its area. Then you rotate it $90^\text{o}$ in the $xy$-plane. Finally, you rotate the loop $60^\text{o}$ in the $yz$-plane. What happens to the magnetic flux through the loop during these steps? Suppose you have a magnetic field directed in the $-\hat{z}$-direction, into the page. There is a flexible, circular loop situated on the page, in the $xy$-plane. You stretch it out in the $\pm x$-direction like a rubber band to change its area. Then you rotate it $90^\text{o}$ in the $xy$-plane. Finally, you rotate the loop $60^\text{o}$ in the $yz$-plane. What happens to the magnetic flux through the loop during these steps?
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   * We represent the steps with the following visual:   * We represent the steps with the following visual:
  
-{{ 184_notes:12_steps_for_loop.png?1000 |Steps}}+[{{ 184_notes:12_steps_for_loop.png?1000 |Steps}}]
 ====Solution==== ====Solution====
 Since the magnetic field has a uniform direction, and the area of the loop is flat (meaning $d\vec{A}$ does not change direction if we move along the area), then we can simplify the dot product: Since the magnetic field has a uniform direction, and the area of the loop is flat (meaning $d\vec{A}$ does not change direction if we move along the area), then we can simplify the dot product:
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 If the loop were to continue rotating in the last step, eventually we would have zero magnetic flux, and as it rotates back around the other way, we could imagine that the flux would then be defined as "negative", since $\cos \theta$ would become negative -- as long as we don't flip the direction of the area-vector. If the loop were to continue rotating in the last step, eventually we would have zero magnetic flux, and as it rotates back around the other way, we could imagine that the flux would then be defined as "negative", since $\cos \theta$ would become negative -- as long as we don't flip the direction of the area-vector.
  
-{{youtube>gzolf12OYzE?large}}+
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  • Last modified: 2018/04/11 19:38
  • by pwirving