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IBY
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Homework Statement
andHomework Equations
I am trying to get from:
[tex]I_{max}=\sqrt{2}I=\frac{V}{\sqrt{R^2+(\omega L-\frac{1}{\omega C})^2}}[/tex]
to:
[tex]\Delta\omega=\omega_1-\omega_2=\frac{R}{L}[/tex]
The Attempt at a Solution
From the equation above:
[tex]\sqrt{R^2+(\omega L-\frac{1}{\omega C})^2}=\frac{V}{\sqrt{2}I}[/tex]
Square the above:
[tex]R^2+(\omega L-\frac{1}{\omega C})^2=\frac{V^2}{2I^2}[/tex]
Subtract from both sides R squared and square root:
[tex]\omega L-\frac{1}{\omega C}=\sqrt{\frac{V^2}{2 I^2}-R^2}[/tex]
Now, I want to put it in the form of quadratic equation, so I multiply omega:
[tex]\omega^2 L-\frac{1}{C}=\omega \sqrt{\frac{V^2}{2 I^2}-R^2}[/tex]
Now I am stuck. It is not like I can turn V/I=R because that I there is root square mean current, and even if I did, I would end up with an imaginary term. The answer is not imaginary, nor complex.
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