184_notes:examples:week14_ac_graph

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184_notes:examples:week14_ac_graph [2017/11/28 16:34] – created tallpaul184_notes:examples:week14_ac_graph [2017/11/28 16:43] – [Changing Current Induces Voltage in Rectangular Loop] tallpaul
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-===== Changing Current Induces Voltage in Rectangular Loop ===== +===== Analyzing an Alternating Current Graph ===== 
-Suppose you have an increasing current through a long wire, $I(t) = I_0 \frac{t}{t_0}$. Next to this wire, there is a rectangular loop of width $w$ and height $h$. The side of the rectangle is aligned parallel to the wire so that the rectangle is a distance $d$ from the wire, and they are both in the same plane. What is the induced voltage in the rectangle? In what direction is the induced current in the rectangle?+Suppose you are given the following graph of current over time. You can see that the first peak is at the point where $t=0.01\text{ s}$, and $I=0.3\text{ A}$. The graph is shown below. Find the amplitude, period, and frequency of the current, and give an equation that describes the alternating current. 
 + 
 +{{ 184_notes:14_ac_graph_given.png?500 |Graph of Alternating Current}}
  
 ===Facts=== ===Facts===
-  * The current in the long wire increases with time and is $I(t) = I_0 \frac{t}{t_0}$. +  * The first peak in the graph is at $(t=0.01\text{ s}, I=0.3\text{ A})$.
-  * The rectangle has dimensions $w$ by $h$, and a side with length $h$ is parallel to the wire. +
-  * The rectangle and the wire lie in the same plane, and are separated by a distance $d$.+
  
 ===Lacking=== ===Lacking===
-  * $V_{ind}$. +  * Amplitude, period, frequency. 
-  * Direction of $I_{ind}$.+  * Equation for alternating current.
  
 ===Approximations & Assumptions=== ===Approximations & Assumptions===
-  * The long wire is infinitely long and thin and straight. +  * The graph of the current is a sine wave. 
-  * There are no external contributions to the B-field.+  * As indicated on the graph, at $t=0$, we have $I=0$. 
 +  * The current is centered about $I=0$, that is, the sine wave has not been shifted vertically at all.
  
 ===Representations=== ===Representations===
-  * We represent the magnetic field from a very long straight wire as $$B = \frac{\mu_0 I}{2 \pi r}$$ where direction is determined based on the right hand rule +  * We represent the graph as given in the example statement.
-  * We represent magnetic flux as $$\Phi_B = \int \vec{B} \bullet \text{d}\vec{A}$$ +
-  * We can represent induced voltage as $$V_{ind} = -\frac{\text{d}\Phi}{\text{d}t}$$ +
-  * We represent the situation with the following visual. We arbitrarily choose a direction for the current. +
- +
-{{ 184_notes:14_wire_rectangle.png?500 |Wire and Rectangle}}+
 ====Solution==== ====Solution====
 In order to find the induced voltage, we will need the magnetic flux. This requires defining an area-vector and determining the magnetic field. We can use the right hand rule to determine the the magnetic field from the wire is into the page ($-\hat{z}$) near the rectangle. For convenience, we will also define for the area vector to be into the page. Since they both point in the same direction, the dot product simplifies: In order to find the induced voltage, we will need the magnetic flux. This requires defining an area-vector and determining the magnetic field. We can use the right hand rule to determine the the magnetic field from the wire is into the page ($-\hat{z}$) near the rectangle. For convenience, we will also define for the area vector to be into the page. Since they both point in the same direction, the dot product simplifies:
  • 184_notes/examples/week14_ac_graph.txt
  • Last modified: 2018/08/09 19:19
  • by curdemma