# Minimize Impedance - (eletromagnetism)

1. Apr 24, 2005

OK, I think I am doing this question right, but I'm not exactly sure. The question is as follows:

For an RLC circuit with a resistance of $$16k\ohm$$, a capacitance of $$8.0\mu F$$ and an inductance of $$38.0H$$ what frequency is needed to minimize the impedance?

Well impedance is give by:
$$Z = \sqrt{R^2 + (X_C - X_L)^2}$$

Putting $$X_C$$ and $$X_L$$ in terms of $$L, C, \omega$$ we then have:

$$Z =\sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$$
Minimum impedance is acheived at resonance, so $$Z = R$$

Thus we have:
$$R =\sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$$

Solving this for $$\omega$$ yields:

$$\omega = \frac{1}{\sqrt{LC}}$$

And frequency is given by: $$f = \frac{\omega}{2\pi}$$

So solving $$f$$ for $$\omega$$ and substituting into the equation above gives:

$$f 2\pi = \frac{1}{\sqrt{LC}}$$

Now solving for $$f$$ yields:
$$f = \frac{1}{2\pi\sqrt{LC}}$$

And finally plugging in $$L,\,C$$ from above gives:

$$f = \frac{1}{2\pi\sqrt{(38.0H)(8.0\mu F)}} = 9.12Hz = 0.009kHz$$

So I'm pretty sure there are going to be a few questions like this on my test tomorrow, so I just want to make sure I'm doing this correctly. Thank you.

2. Apr 24, 2005

Looks good.