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A circuit containing a capacitor and an inductor in series has a resonant frequency of 575 kilohertz.
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The inductor in the circuit has an inductance of 1.25 henries.
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What is the capacitance of the capacitor?
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Give your answer in scientific notation to two decimal places.
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Letโs say that this is the circuit weโre working with.
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It has a capacitor and inductor.
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And since this is an alternating current circuit, it has a variable voltage supply.
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Weโre told that the resonant frequency of this circuit, weโll call it ๐ sub ๐
, is 575 kilohertz.
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Thatโs 575,000 hertz.
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Weโre also told that the inductance of our inductor is 1.25 henries.
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The henry is the standard SI unit of inductance.
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Knowing all this, we want to solve for the capacitance of the capacitor.
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Weโre going to do this assuming that our circuit is indeed at its resonant frequency.
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This is the frequency at which the overall opposition to charge flow in the circuit is minimized.
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A circuitโs resonant frequency is achieved when what is called the reactance of the capacitor and the reactance of the inductor are equal.
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All that to say, the resonant frequency in a circuit depends on its capacitance and its inductance.
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These three quantities are related through this equation.
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The resonant frequency of a circuit equals one over two ๐ times the square root of its inductance times its capacitance.
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Note that in our scenario, we know the resonant frequency of our circuit and its inductance.
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Itโs the capacitance we want to solve for.
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So letโs rearrange this resonant frequency equation to solve for ๐ถ, the capacitance.
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If we multiply both sides of this equation by the square root of ๐ถ over ๐ sub ๐
, then on the left-hand side, that resonant frequency cancels out.
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And on the right, the square root of capacitance cancels.
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We therefore have that the square root of ๐ถ equals one over two ๐ ๐ sub ๐
times the square root of ๐ฟ.
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Then, if we square both sides, we get this equation for the capacitance: one over four ๐ squared times ๐ sub ๐
squared times ๐ฟ.
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Letโs now recall that we know the resonant frequency of our circuit as well as its inductance.
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We then substitute those values into this expression: 575 times 10 to the third hertz for the resonant frequency and 1.25 henries for the inductance.
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When we enter this expression on our calculator, we find a result in scientific notation to two decimal places of 6.13 times 10 to the negative 14th farads.
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This is the capacitance of the capacitor in our circuit.