WEBVTT
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What is the resonant frequency of the circuit shown in the diagram?
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The circuit consists of an alternating-voltage source connected to a series combination of a 35-Ξ© resistor, a 7.5-henry inductor, and a 350-microfarad capacitor.
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And weβre asked to find the resonant frequency of this circuit.
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Recall that the inductive reactance in a circuit is the angular frequency of the voltage source times the inductance.
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And the capacitive reactance is one divided by the angular frequency of the voltage source times the capacitance.
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On resonance, these two reactances are equal.
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If we call the resonant angular frequency π naught, then we have that π naught πΏ is equal to one over π naught πΆ, which we can solve for π naught.
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When we solve this equation for π naught, we find that the resonant angular frequency is equal to one divided by the square root of the inductance of the inductor times the capacitance of the capacitor.
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Now, this is a formula for angular frequency, but weβre looking for just regular frequency.
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So we need to use the relationship that angular frequency is two π times the regular frequency.
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Alright, so letβs plug our definition for angular frequency into our equation for the resonant angular frequency.
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We have two times π times the resonant frequency is equal to one divided by the square root of the inductance times the capacitance.
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To get this expression into the final form we need, we simply divide both sides by two π.
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On the left-hand side, two π divided by two π is one, and weβre just left with π naught.
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On the right-hand side, the two π just becomes part of the denominator of our fraction.
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This leaves us with the final formula we need.
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Resonant frequency is equal to one divided by two π times the square root of the inductance, in henries, times the capacitance, in farads.
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So now we just need to plug in values.
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We have an inductance in henries.
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Itβs 7.5 henries.
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However, our capacitance is given in microfarads instead of farads.
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To convert to farads, recall that there are one million microfarads per farad.
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In other words, one microfarad is equivalent to 10 to the negative sixth farads.
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Since we have 350 microfarads, our capacitance is equivalent to 350 times 10 to the negative sixth farads.
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Plugging our inductance and capacitance into our formula for resonant frequency, this gives us one divided by two π times the square root of 350 times 10 to the negative sixth farads times 7.5 henries.
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It turns out that the square root of one farad times one henry is one second.
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So we can rewrite the denominator with units of seconds.
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Now, one divided by seconds is the unit hertz, which is used for frequency.
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So now we have an expression for the resonant frequency.
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Thatβs a number times a unit hertz, which is the right unit for frequency.
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So now all we have to do is evaluate this number with a calculator.
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When we do this evaluation, we find that the entire numerical expression is approximately equal to 3.1.
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So the resonant frequency of this circuit is 3.1 hertz.
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Itβs worth noting that the 35-Ξ© resistor played no role in our calculation of the resonant frequency.