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--- 184_notes:examples:week12_decreasing_flux [2017/11/10 14:59] – created tallpaul
+++ 184_notes:examples:week12_decreasing_flux [2018/08/09 18:46] (current) – curdemma
@@ Line 1: / Line 1: @@
+[[184_notes:ind_i|Return to The Curly Electric Field and Induced Current notes]]
 ===== Decreasing Flux =====
-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?
+Say we have a bar that is sliding down a pair of connected conductive rails (so current is free to flow through the loop created by the bar and rails), which is sitting in a magnetic field that points into the page. If the bar slides along the rails to decrease the area of the loop, what happens?
 ===Facts===
   * The magnetic field is directed into the page.
-  * The steps for changing and rotating the loop are outlined in the problem statement.
+  * The loop is oriented so that the magnetic flux is nonzero.
 ===Lacking===
-  * A description of the magnetic flux.
+  * A description of the magnetic flux and/or induced current.
 ===Approximations & Assumptions===
   * The magnetic field is constant in time, and the same everywhere.
-  * The steps for changing and rotating the loop happen in a reasonable enough amount of time to describe the flux as the motions are happening.
+  * The sliding of the bar along the rails happens in a reasonable enough amount of time to describe the flux as the motion is happening.
 ===Representations===
@@ Line 18: / Line 20: @@
   * We represent the steps with the following visual:
-{{ 184_notes:12_steps_for_loop.png?1000 |Steps}}
+[{{ 184_notes:12_rail_flux.png?400 |Flux Through Loop}}]
 ====Solution====
 Since the magnetic field has a uniform direction, and the area of the loop is flat (meaning $\text{d}\vec{A}$ does not change direction if we move along the area), then we can simplify the dot product:
@@ Line 27: / Line 29: @@
 $$\int B\text{d}A\cos\theta =B\cos\theta \int \text{d}A = BA\cos\theta$$
-It will be easier to concern ourselves with this value, rather than try to describe the integral calculation each time. At the beginning of the motion, the loop is just a circle. Its area vector and the magnetic field are aligned (parallel), so it has some nonzero magnetic flux.
+It will be easier to concern ourselves with this value, rather than try to describe the integral calculation each time. As the bar begins moving to the left, the area within the loop begins to close up, as shown below:
+[{{ 184_notes:12_rail_flux_decreased.png?400 |Decreased Flux}}]
+It should be easy to see the that the magnetic field and the orientation of the loop are not changing, but the area of the loop is decreasing. **The flux through the loop is therefore decreasing, which indicates that a current is induced in the loop**.
+To determine the direction of the induced current, consider that when we say the flux decreases, this carries the //__assumption that the area vector points in the same direction as the magnetic field__//, which is into the page. This helps us define what we mean by "positive", "negative", "increase", and "decrease". A positive flux then means that the area vector and magnetic field point in the same direction. The fact that our flux is decreasing means that initially there was a large positive magnetic flux and at some time later there was a smaller, positive magnetic flux.
-**Step 1:** As soon as we begin to stretch out our circle, we can imagine that its area begins to decrease, much like when you pinch a straw. We don't change its orientation with respect to the magnetic field, but since its area decreases, we expect that the flux through the loop will also decrease.
+Based on Faraday's law, we know that the induced current should create a magnetic field that opposes the change in flux. The change in flux for this example is a "decrease", going from a positive value toward zero (or towards a "negative" flux). So we expect the induced current to create a "positive" magnetic field, where "positive" here means that it is pointing in the same direction as the area-vector. Since we chose the area vector to point into the page, we expect for the magnetic field from the induced current to also point into the page. If we point our thumb in the direction of the magnetic field from the induced current then curl our finger, we find that **the induced current is clockwise around the loop**.
-**Step 2:** As we rotate the stretched loop, notice that the area vector and magnetic field remain perfectly aligned ($\theta$ does not change). Further, the area itself is not changing. Since the magnetic field is also constant, we don't expect the flux the change at all during this rotation. It remains the same!
+Alternatively, you could think of the mobile charges inside the moving bar. Since the bar is moving inside a magnetic field, mobile charges will experience an acceleration due to the magnetic force. The force is $\vec{F} = q \vec{v} \times \vec{B}$, which points down for positive charges, implying that conventional current will be clockwise, which is the same result we reached using Faraday's law. The result is shown below.
-**Step 3:** As we rotate the stretched loop again, we are rotating it in such a way that the area vector also rotates. In fact, the area vector becomes less and less aligned with the magnetic field, which indicates that $\cos \theta$ will be decreasing during this motion. This causes us to expect that the magnetic flux through the loop will decrease during this rotation. Alternatively, based on the perspective shown in the representation, one can imagine that the magnetic field "sees" less and less of the loop, indicating that the flux is decreasing.
+[{{ 184_notes:12_rail_current.png?300 |Induced Current}}]
-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>lGpZ63wuroY?large}}