Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 35 - AC Circuits - Exercises and Problems - Page 1055: 65

Answer

See the detailed answer below.

Work Step by Step

We know that a resistor's voltage and current are in phase while the current leads the voltage of a capacitor by 90$^\circ$. And we are given that the transformer voltage produces a 6.0 V rms output that leads the input voltage by 45$^\circ$. This means that we need to make $V_R=V_C$ to make $V_R$ and $I$ leads the input voltage ($\varepsilon$) by $45^\circ$. $$V_R=V_C\\ \color{red}{\bf\not} IR= \color{red}{\bf\not} IX_C\\ R=X_C$$ $$R=\dfrac{1}{2\pi f C}\tag 1$$ Thus, the phase angle is then $$\phi=\tan^{-1}\left[ \dfrac{|X_L-X_C|}{R}\right]=\tan^{-1}\left[ \dfrac{X_C}{R}\right]=\bf 45^\circ$$ See the first graph below of the phasor diagram for an $RC$ circuit. We designed the circuit as shown below in the second figure, so we need to make sure that the output voltage is 6 V. $$(V_R)_{\rm rms}=I_{\rm rms}R=\dfrac{\varepsilon_{\rm rms}R}{\sqrt{X_C^2+R^2}}$$ where $X_C=R$, so $$(V_R)_{\rm rms}=I_{\rm rms}R=\dfrac{\varepsilon_{\rm rms} }{\sqrt{2}}=\dfrac{12}{\sqrt2}=\bf 8.49\;\rm V\gt 6\;\rm V$$ which is greater than the needed output voltage of 6 V. Now we can reduce the output voltage by using two in-series resistors that have a net resistance as the original resistor $R$, as shown in the third figure below; where $R_1=R_2=R$. So, $V_1=IR_1$, and $V_2=IR_2$ So, $$\dfrac{V_2}{V_R}=\dfrac{V_2}{V_1+V_2}=\dfrac{R_2}{R_1+R_2}=\dfrac{R_2}{R}$$ Hence, $$R_2=\dfrac{RV_2}{V_R}$$ where $V_2$ is assumed to be exactly 6 V while $V_R=12/\sqrt2$, $$R_2=\dfrac{ 6\sqrt2}{12}R$$ $$\boxed{R_2= 0.7071R}$$ And hence, $$\boxed{R_1= 0.2929R}$$ We can see that any values that obey these formulas will give us the needed circuit.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.