Question #208288

Two capacitors with capacitances of C1 =0.3 μF and C2=0.2 μF are

connected in series to an alternating current circuit with a voltage of 300

V and a frequency of 50 Hz. Find: (1) the intensity of the current in the

circuit, (2) the potential drop across the first and second capacitors.


1
Expert's answer
2021-06-21T11:38:00-0400

(1) Let's first find the equivalent capacitance of two capacitors connected in series:


1Ceq=1C1+1C2,\dfrac{1}{C_{eq}}=\dfrac{1}{C_{1}}+\dfrac{1}{C_{2}},Ceq=11C1+1C2=110.3 μF+10.2 μF=0.12106 F.C_{eq}=\dfrac{1}{\dfrac{1}{C_{1}}+\dfrac{1}{C_{2}}}=\dfrac{1}{\dfrac{1}{0.3\ \mu F}+\dfrac{1}{0.2\ \mu F}}=0.12\cdot10^{-6}\ F.

Then, we can find the capacitive reactance as follows:


XC=12πfCeq=12π50 Hz0.12106 F=2.65104 Ω.X_C=\dfrac{1}{2\pi fC_{eq}}=\dfrac{1}{2\pi\cdot50\ Hz\cdot0.12\cdot10^{-6}\ F}=2.65\cdot10^4\ \Omega.

Finally, we can find the intensity of the current in the circuit:


I=VXC=300 V2.65104 Ω=1.1102 A.I=\dfrac{V}{X_C}=\dfrac{300\ V}{2.65\cdot10^4\ \Omega}=1.1\cdot10^{-2}\ A.

(2) Let's first find the charge on the equivalent capacitor:


Q=CeqΔV=0.12106 F300 V=3.6105 C.Q=C_{eq}\Delta V=0.12\cdot10^{-6}\ F\cdot300\ V=3.6\cdot10^{-5}\ C.

For the series connection of the capacitors, the magnitude of the charge will be the same, therefore, we can write:


Q1=Q2=Q.Q_1=Q_2=Q.

Finally, we can find the potential drop across the first and second capacitors:


ΔV1=QC1=3.6105 C0.3106 F=120 V,\Delta V_1=\dfrac{Q}{C_1}=\dfrac{3.6\cdot10^{-5}\ C}{0.3\cdot10^{-6}\ F}=120\ V,ΔV2=QC2=3.6105 C0.2106 F=180 V.\Delta V_2=\dfrac{Q}{C_2}=\dfrac{3.6\cdot10^{-5}\ C}{0.2\cdot10^{-6}\ F}=180\ V.

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