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Answer :
To find the ratios XL/R and XC/R in a series RLC circuit at 5 times the resonant frequency, we must consider the relationship between impedance, inductive reactance, and capacitive reactance given that impedance is 5 times greater at this frequency. Using algebraic manipulation of the impedance formula Z = \/R^2 + (XL - XC)^2 at 5 times the resonant frequency, we can obtain the desired ratios.
The question seeks to determine the ratios of inductive reactance (XL) and capacitive reactance (XC) to resistance (R) in a series RLC circuit when the frequency is 5 times the resonant frequency (f0). At resonance, the impedance is purely resistive (Z = R), and the inductive and capacitive reactances are equal (XL = XC). If the impedance at 5f0 is 5 times the resonant impedance, we have Z = 5R. Since Z = \/R^2 + (XL - XC)^2 and XL = XC at f0, at 5f0 the impedance becomes Z = \/R^2 + ((
5wL)^2 - (1/5wC)^2).
To find XL/R and XC/R, we use the definition of reactances where w is the angular frequency (w = 2πf). We have XL = wL and XC = 1/wC. At 5f0, w is 5 times the angular frequency at resonance (w0), hence XL at 5f0 is
5w0L, and XC at 5f0 is 1/5w0C.
We can use the given information to find XL/R and XC/R. Let's denote w0L (the inductive reactance at resonance) as XLR and 1/w0C (the capacitive reactance at resonance) as XCR. Then XL/R at 5f0 would be 5XLR/R and XC/R at 5f0 would be XCR/5R. As we have Z = 5R at 5f0, this leads to 25XLR^2/R^2 - XCR^2/25R^2 = 24R^2, which gives us the ratios of XL/R and XC/R after some algebraic manipulation.
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