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+ | Section 22.2 in Matter and Interactions (4th edition) | ||
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+ | [[184_notes: | ||
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===== Changing Flux from an Alternating Current ===== | ===== Changing Flux from an Alternating Current ===== | ||
As we said before, one of the most important sources of a changing magnetic field is an alternating current. This is what actually comes out of the wall outlets; as opposed to the current from a battery which is a constant current (or a direct current). We are only briefly going to talk about alternating current as it refers to induction and changing magnetic flux, but there are many more applications of alternating current, especially with regard to circuits, resistors, and capacitors. For the purposes of these notes, we will talk about how we represent an alternating current, how that alternating current can produce an induced current/ | As we said before, one of the most important sources of a changing magnetic field is an alternating current. This is what actually comes out of the wall outlets; as opposed to the current from a battery which is a constant current (or a direct current). We are only briefly going to talk about alternating current as it refers to induction and changing magnetic flux, but there are many more applications of alternating current, especially with regard to circuits, resistors, and capacitors. For the purposes of these notes, we will talk about how we represent an alternating current, how that alternating current can produce an induced current/ | ||
- | ==== Alternating Current ==== | + | ===== Alternating Current |
- | FIXME Add Figure 14.6 | + | [{{ 184_notes: |
When a current is constantly switching between positive and negative values, we call this an **alternating current**. Typically, this means that we have a current that is represented by a sine graph (shown in the figure to the right), though there can be other types of alternating current that look like step functions (also called [[https:// | When a current is constantly switching between positive and negative values, we call this an **alternating current**. Typically, this means that we have a current that is represented by a sine graph (shown in the figure to the right), though there can be other types of alternating current that look like step functions (also called [[https:// | ||
- | $$I=I_0sin(2\pi \cdot f \cdot t)$$ | + | $$I=I_0\sin(2\pi \cdot f \cdot t)$$ |
where $I$ is the current at any particular time $t$, $I_0$ is the //maximum// current (this is a constant value), and $f$ is the frequency, which in general tells you how fast the current is changing between positive and negative with units of $\frac{1}{s}$ or $Hz$ (Hertz). More specifically, | where $I$ is the current at any particular time $t$, $I_0$ is the //maximum// current (this is a constant value), and $f$ is the frequency, which in general tells you how fast the current is changing between positive and negative with units of $\frac{1}{s}$ or $Hz$ (Hertz). More specifically, | ||
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This means that a larger period would be related to a smaller frequency, and a smaller period would be related to a higher frequency. Since the period is often easier to think about conceptually, | This means that a larger period would be related to a smaller frequency, and a smaller period would be related to a higher frequency. Since the period is often easier to think about conceptually, | ||
- | ==== Voltage Transformer ==== | + | ===== Voltage Transformer |
If you have an oscillating current, this would also mean that you would have an oscillating magnetic field everywhere around the wire (since currents create magnetic fields). If there is an oscillating magnetic field, this means that there will also be an induced potential/ | If you have an oscillating current, this would also mean that you would have an oscillating magnetic field everywhere around the wire (since currents create magnetic fields). If there is an oscillating magnetic field, this means that there will also be an induced potential/ | ||
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+ | [{{184_notes: | ||
As you figured out in the project last week, a rotating loop in a magnetic field will create a oscillating current and oscillating voltage. This idea is exactly what happens in a power generator, just on a much larger scale. In a power generator there are thousands of loops, rotating very quickly in a large magnetic field to produce a very large current with a frequency of 60 Hz (in the U.S. at least). However, because power (or the energy transferred to heat) is related to $P=I^2R$, sending a very large current over large distances (like from the generator to your house) would result in a large loss of power over that wire. This is a problem because any power lost on the wires can't be used in your house and if that power is large enough it could heat up the wires to point where they are damaged. | As you figured out in the project last week, a rotating loop in a magnetic field will create a oscillating current and oscillating voltage. This idea is exactly what happens in a power generator, just on a much larger scale. In a power generator there are thousands of loops, rotating very quickly in a large magnetic field to produce a very large current with a frequency of 60 Hz (in the U.S. at least). However, because power (or the energy transferred to heat) is related to $P=I^2R$, sending a very large current over large distances (like from the generator to your house) would result in a large loss of power over that wire. This is a problem because any power lost on the wires can't be used in your house and if that power is large enough it could heat up the wires to point where they are damaged. | ||
- | To get around this problem, a step up transformer is used to change a low voltage, high current circuit (like what comes out the generator) into a high voltage, low current circuit for transport from the generator to the neighborhoods or wherever it is needed. A step down transformer is then used close to the neighborhoods to return the high voltage, low current back to a low voltage, high current circuit that is then used in your house. You may have seen these around your neighborhood - they look like small boxes attached to the power lines overhead, generally on the lines going from a larger street into a residential area.) | + | To get around this problem, a step up transformer is used to change a low voltage, high current circuit (like what comes out the generator) into a high voltage, low current circuit for transport from the generator to the neighborhoods or wherever it is needed. A step down transformer is then used close to the neighborhoods to return the high voltage, low current back to a low voltage, high current circuit that is then used in your house. You may have seen these around your neighborhood - they look like small boxes attached to the power lines overhead, generally on the lines going from a larger street into a residential area (shown in the figure to the left). |
- | In these notes, we will go through how a step up transformer works and how it uses induction to change the voltage from a low voltage to a high voltage. We will use a basic transformer, | + | [{{ 184_notes: |
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+ | In these notes, we will go through how a step up transformer works and how it uses induction to change the voltage from a low voltage to a high voltage. We will use a basic transformer, | ||
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+ | [{{184_notes: | ||
- | For this problem it is easy to overcomplicate | + | Because there is an oscillating potential/ |
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+ | [{{ 184_notes: | ||
- | **Step 1**: There is an oscillating potential/ | + | If we put the secondary |
- | $$V_1=V_0 sin(\omega t)$$ | + | $$-V_{S}=\frac{d\Phi_{B_{S}}}{dt}$$ |
- | **Step 2**: If you pick a material (like iron) that easily responds to the magnetic field, then this changing | + | We can then rewrite the flux $\Phi_{B2}$ using the flux definition: |
- | $$B_1=B_2$$ | + | $$\Phi_{B_{S}}=\int \vec{B}_S \bullet \vec{dA}_S$$ |
- | $$A_1=A_2$$ | + | where the area here would be the cross-sectional area of the iron ring (since this is where the magnetic field is). The direction of the magnetic field would always be perpendicular to the area of the cylinder - so the flux would simplfy to: |
- | Note that $B_2$ here is the magnetic | + | $$\Phi_{B_{S}}=B_S A_S$$ |
+ | But this is the magnetic | ||
+ | $$\Phi_{B_{S}}=B_S A_S N_S$$ | ||
+ | We can then plug this into the voltage equation we wrote above: | ||
+ | $$-V_{S}=\frac{d}{dt}\biggl(B_S A_S N_S\biggr)$$ | ||
- | **Step 3**: If we put the second solenoid on the end of the bar, this changing | + | Since $N_S$ and $A_S$ are constant with respect to time (not add/taking away loops or increasing/ |
- | $$-V_{2}=\frac{d\Phi_{B2}}{dt}$$ | + | $$-V_{S}=N_S A_S \frac{d}{dt}\biggl(B_S\biggr)$$ |
+ | // | ||
+ | $$-V_{S}=N_S A_P \frac{d}{dt}\biggl(B_P\biggr)$$ | ||
- | **Step 4**: We can rewrite | + | If we look at the $A_P\frac{d}{dt}\biggl(B_P\biggr)$ term, this looks very much like the changing |
- | $$\Phi_{B2}=\int \vec{B}_2 \bullet \vec{dA}_2$$ | + | $$\frac{d\Phi_{B_{P}}}{dt}=\frac{d}{dt}\biggl(B_P A_P\biggr)$$ |
- | The direction of the magnetic | + | If we use Faraday' |
- | $$\Phi_{B2}=B_2A_2$$ | + | $$-V_P=N_P |
- | But this is the magnetic | + | $$\frac{d}{dt}\biggl(B_P A_P\biggr)=\frac{-V_P}{N_P}$$ |
- | $$\Phi_{B2}=B_2A_2N_2$$ | + | |
- | We can then plug this into the equation we wrote in Step 2: | + | |
- | $$-V_{2}=\frac{d}{dt}\biggl(B_2A_2N_2\biggr)$$ | + | |
- | **Step 5**: $N_2$ and $A_2$ are constant with respect | + | Finally, We can plug this result into the $V_S$ equation above to get the potential in the secondary solenoid as it relates to the potential in the primary solenoid: |
- | $$-V_{2}=N_2A_2\frac{d}{dt}\biggl(B_2\biggr)$$ | + | $$-V_S = N_S \frac{-V_P}{N_P}$$ |
- | And we can use what we found from Step 2 to rewrite the flux in terms of source B-field and the source area: | + | $$V_S=V_P \frac{N_S}{N_P}$$ |
- | $$-V_{2}=N_2A_1\frac{d}{dt}\biggl(B_1\biggr)$$ | + | |
- | **Step 6**: If we look at the $A_1\frac{d}{dt}\biggl(B_1\biggr)$ term, this looks very much like the changing magnetic flux (or induced voltage) through a single loop of the //first// solenoid: | + | This equation then tells us that that if we want the voltage in the secondary solenoid to be 100 times greater than the voltage in the primary solenoid, the secondary solenoid must have 100 times as many loops as the primary solenoid! Remember that even though there are not explicit sines or cosines in our equation, this whole process hinges on the fact that the current, and thus magnetic field and voltage, are oscillating. You would not be able to make a voltage transformer in the same way using a direct current because there would no longer be a changing magnetic flux. (This was one of the [[https://energy.gov/articles/war-currents-ac-vs-dc-power|biggest arguments made back in the 1880s]] for using alternating current |
- | $$\frac{d\Phi_{B1}}{dt}=\frac{d}{dt}\biggl(B_1 A_1\biggr)$$ | + | |
- | If we use Faraday' | + | |
- | $$-V_1=N_1 \frac{d}{dt}\biggl(B_1 A_1\biggr)$$ | + | |
- | $$\frac{d}{dt}\biggl(B_1 A_1\biggr)=\frac{-V_1}{N_1}$$ | + | |
- | **Step | + | ==== Examples ==== |
- | $$-V_2=N_2\frac{-V_1}{N_1}$$ | + | |
- | $$V_2=\frac{V_1 N_2}{N_1}$$ | + | |
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+ | {{youtube> | ||
- | Since we want an increase in potential by a factor of 10, this equation then tells that the second solenoid needs to have a 10 times as many coils as the first solenoid. So they could pick either the 10 and 100 turn solenoids or the 20 and 200 turn solenoids. Note, you do not really need to do any derivatives or flux calculations in this problem! |