Blog: An ac generator supplies an rms voltage of 5.00 V to an RC circuit. At the frequency of 20.0 kHz, the rms current in the circuit is 38.0 mA;...

Saturday, May 17, 2014

An ac generator supplies an rms voltage of 5.00 V to an RC circuit. At the frequency of 20.0 kHz, the rms current in the circuit is 38.0 mA;...

The relationship between the voltage and current in AC circuit can be written in the form analogous to Ohm's Law:

$\displaystyle{V}={I}{Z}$

, where I and V are effective, or rms, values of the current and voltage, and Z is the impedance. For the RC circuit, the impedance is

$\displaystyle{Z}=\sqrt{{{{R}}^{{2}}+{{X}_{{C}}^{{2}}}}}$

, where $\displaystyle{X}_{{C}}=\frac{{1}}{{\omega{C}}}$ .

Since we don't know the values of R and C (and therefore ), we have to use the...

The relationship between the voltage and current in AC circuit can be written in the form analogous to Ohm's Law:

$\displaystyle{V}={I}{Z}$

, where I and V are effective, or rms, values of the current and voltage, and Z is the impedance. For the RC circuit, the impedance is

$\displaystyle{Z}=\sqrt{{{{R}}^{{2}}+{{X}_{{C}}^{{2}}}}}$

, where $\displaystyle{X}_{{C}}=\frac{{1}}{{\omega{C}}}$ .

Since we don't know the values of R and C (and therefore ), we have to use the known values of V, I and and write the system of two equations with two variables. It is easier to use the Ohm's Law with the both sides squared:

$\displaystyle{{V}}^{{2}}={{I}}^{{2}}{\left({{R}}^{{2}}+{{X}_{{c}}^{{2}}}\right)}$

For the frequency f = 20 kHz, the angular frequency is

$\displaystyle{w}={2}\pi{f{=}}{126}\cdot{{10}}^{{3}}\frac{{{r}{a}{d}}}{{s}}$

and the current is $\displaystyle{I}={38}\cdot{{10}}^{{-{3}}}{A}$ .So the equation becomes

$\displaystyle{{5}}^{{2}}={{\left({38}\cdot{{10}}^{{-{3}}}\right)}}^{{2}}{\left({{R}}^{{2}}+\frac{{1}}{{{{\left({126}\cdot{{10}}^{{3}}\right)}}^{{2}}\cdot{{C}}^{{2}}}}\right)}$

Divide by the coefficient on the right side in order to isolate the parenthesis:

$\displaystyle{0.017}\cdot{{10}}^{{6}}={{R}}^{{2}}+\frac{{1}}{{{15},{876}\cdot{{10}}^{{6}}\cdot{{C}}^{{2}}}}$

Similarly, for the frequency f = 28 kHz, the angular frequency is

$\displaystyle{w}={2}\pi{f{=}}{176}\cdot{{10}}^{{3}}\frac{{{r}{a}{d}}}{{s}}$ and the equation becomes, after plugging in the current of 50 mA:

$\displaystyle{{5}}^{{2}}={{\left({50}\cdot{{10}}^{{-{3}}}\right)}}^{{2}}{\left({{R}}^{{2}}+\frac{{1}}{{{{\left({176}\cdot{{10}}^{{3}}\right)}}^{{2}}\cdot{{C}}^{{2}}}}\right)}$

This becomes, after dividing by the coefficient in front of the parenthesis

$\displaystyle{0.01}\cdot{{10}}^{{6}}={{R}}^{{2}}+\frac{{1}}{{{30},{976}\cdot{{10}}^{{6}}\cdot{{C}}^{{2}}}}$

So we have two equations with two unknown variables, R and C. We can solve it by eliminating R. Subtract the second equation from the first one. $\displaystyle{{R}}^{{2}}$ will cancel out and we will get

$\displaystyle{0.007}\cdot{{10}}^{{6}}=\frac{{1}}{{{C}}^{{2}}}{\left(\frac{{1}}{{15876}}-\frac{{1}}{{30976}}\right)}\cdot{{10}}^{{-{6}}}$

$\displaystyle{7}\cdot{{10}}^{{3}}=\frac{{1}}{{{C}}^{{2}}}\cdot{3.07}\cdot{{10}}^{{-{5}}}\cdot{{10}}^{{-{6}}}$

Finally, from here $\displaystyle{{C}}^{{2}}=\frac{{{3.07}\cdot{{10}}^{{-{11}}}}}{{{7}\cdot{{10}}^{{3}}}}$

and $\displaystyle{C}={0.66}\cdot{{10}}^{{-{7}}}={6.6}\cdot{{10}}^{{-{8}}}$ Farad.

The resistance then can be found from one of the equations. Using the second equation,

$\displaystyle{{R}}^{{2}}={0.01}\cdot{{10}}^{{6}}-\frac{{1}}{{{30},{976}\cdot{{10}}^{{6}}\cdot{{C}}^{{2}}}}$

Plugging in C results in

$\displaystyle{{R}}^{{2}}={0.01}\cdot{{10}}^{{6}}-{7.4}\cdot{{10}}^{{3}}={10}\cdot{{10}}^{{3}}-{7.4}\cdot{{10}}^{{3}}={2.6}\cdot{{10}}^{{3}}$ $\displaystyle<{b}{r}{d}{a}{t}{a}-{m}{c}{e}-{b}{o}{g{{u}}}{s}=\text{1}>$

$\displaystyle{R}={51}$ Ohm

So the values of R and C are 51 Ohm and 6.6*10^(-8) Farad, respectively.

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Saturday, May 17, 2014

An ac generator supplies an rms voltage of 5.00 V to an RC circuit. At the frequency of 20.0 kHz, the rms current in the circuit is 38.0 mA;...

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What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?

Saturday, May 17, 2014

An ac generator supplies an rms voltage of 5.00 V to an RC circuit. At the frequency of 20.0 kHz, the rms current in the circuit is 38.0 mA;...

No comments:

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What is the Exposition, Rising Action, Climax, and Falling Action of &quot;One Thousand Dollars&quot;?

What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?