An impedance coil whose resistance and inductance are 18 ohms and 3.82 mH, respectively, is connected in parallel with a series combination of a 5.3 uF capacitor and a variable resistor. For what value of resistance will the circuit be in resonance at 1000 cps?
Let series resistor and inductor combination be "R_L+jX_L"
Let series capacitor and resistor combination be "Rc-jXc"
When these are connected in parallel the total impedance should be resistive, that is the condition for resonance.
i. e imaginary term of parallel combination of "R_L+jX_L \\space and\\space Rc-jXc" =0
"i.e (RcX_L - R_LXc) (R_L+Rc) - \\newline (X_L-Xc) (R_LRc+X_LXc) =0"
Simplifying we have,
Substituting given inductor, capacitor resistor and frequency values we have,
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