See picture to see the circuit. Both resistors are worth R.
V(t) is an emf with angular frequency \(\omega\) and amplitude \(V_0\).
1)Find the current that passes through the emf.
2)Let \(V_{AB}\) be worth \(V_B-V_A\). Demonstrate that \(|V_{AB}|^2=V_0^2\) for all \(\omega\).
3)Find the angle difference of phase between the current passing through the emf and the current reaching a capacitor.
My attempt:
\(V(t)=V_0 \cos (\omega t)\). Also, \(V(t)=Zi(t)\) where Z is the impedance of the circuit.
As both branches of the circuit are in parallel, the admittance of both branches sum up. \(G=\frac{1}{Z}\).
So \(G=G_1+G_2 \Rightarrow Z=\frac{1}{G}=\frac{1}{G_1+G_2}=\frac{1}{\left ( \frac{1}{Z_1}+\frac{1}{Z_2} \right )}\) where \(Z_1\) and \(Z_2\) are the impedances of each branch, respectively.
But \(Z_1=Z_2\). Thus \(Z=\frac{Z_1}{2}\).
\(Z_1=R-\frac{1}{i\omega C}\), thus \(Z=\frac{R}{2}+\frac{i}{2\omega C}\).
Finally, \(i(t)=\frac{2V_0 \omega C \cos (\omega t)}{R\omega C+i}\), where the i in the denominator is the complex number.
Am I right for question 1)? Did I make an error?
For 2), I'm not really sure about how to proceed.
I believe that \(V_B-V_A=\frac{i(t)R}{4}-\frac{q(t)}{C}\). But I'm unsure and even if it were right, I don't know how to continue. I know that \(i=\frac{dq(t)}{dt}\), but does this help?
Any help is greatly appreciated! Thanks in advance.
V(t) is an emf with angular frequency \(\omega\) and amplitude \(V_0\).
1)Find the current that passes through the emf.
2)Let \(V_{AB}\) be worth \(V_B-V_A\). Demonstrate that \(|V_{AB}|^2=V_0^2\) for all \(\omega\).
3)Find the angle difference of phase between the current passing through the emf and the current reaching a capacitor.
My attempt:
\(V(t)=V_0 \cos (\omega t)\). Also, \(V(t)=Zi(t)\) where Z is the impedance of the circuit.
As both branches of the circuit are in parallel, the admittance of both branches sum up. \(G=\frac{1}{Z}\).
So \(G=G_1+G_2 \Rightarrow Z=\frac{1}{G}=\frac{1}{G_1+G_2}=\frac{1}{\left ( \frac{1}{Z_1}+\frac{1}{Z_2} \right )}\) where \(Z_1\) and \(Z_2\) are the impedances of each branch, respectively.
But \(Z_1=Z_2\). Thus \(Z=\frac{Z_1}{2}\).
\(Z_1=R-\frac{1}{i\omega C}\), thus \(Z=\frac{R}{2}+\frac{i}{2\omega C}\).
Finally, \(i(t)=\frac{2V_0 \omega C \cos (\omega t)}{R\omega C+i}\), where the i in the denominator is the complex number.
Am I right for question 1)? Did I make an error?
For 2), I'm not really sure about how to proceed.
I believe that \(V_B-V_A=\frac{i(t)R}{4}-\frac{q(t)}{C}\). But I'm unsure and even if it were right, I don't know how to continue. I know that \(i=\frac{dq(t)}{dt}\), but does this help?
Any help is greatly appreciated! Thanks in advance.
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