An R-C circuit is driven by an alternating voltage of amplitude 110 V and frequency w. Define V0 to be amplitude of the voltage across the resistor. The resistance of the resistor is 1000 ohms and the capacitance of the capacitor is 1 microFarad. What is V0 if w=10 rad/s? Any help you could give on how this is worked out would be appreciated. Thanks.
find impedance: Z= 1000-.159j/wC
figure that impedance, magnitude
then Vo=IR=110^2/Z * R
so you know Z magnitude, and R, solve for Vo
To solve this problem, we can use the impedance of the R-C circuit, which is given by the formula:
Z = √(R^2 + 1/(wC)^2),
where R is the resistance, w is the angular frequency, and C is the capacitance.
Given that R = 1000 ohms, w = 10 rad/s, and C = 1 microFarad (1 μF = 10^(-6) F), we can substitute these values into the formula:
Z = √((1000)^2 + 1/((10)(10^(-6)))^2)
= √(10^6 + 1/10^4)
= √(10^6 + 10^(-4))
= √(10^6 + 10^(-6))
= √(10^6(1 + 10^(-6)))
≈ √(10^6)
Since the impedance of a resistor in an R-C circuit is equal to the resistance, the voltage across the resistor is equal to the voltage across the circuit, which is V0.
Therefore, V0 = Z = √(10^6) ≈ 1000 V.
Therefore, when w = 10 rad/s, the amplitude voltage across the resistor is approximately 1000 V (V0 = 1000 V).
To find the amplitude V0 of the voltage across the resistor in an R-C circuit driven by an alternating voltage, we need to understand the behavior of the circuit with respect to the frequency.
In an R-C circuit, the voltage across the resistor and the voltage across the capacitor are out of phase because of the charging and discharging of the capacitor. The amplitude of the voltage across the resistor can be found using Ohm's Law and the impedance of the capacitor.
Here's how you can work it out:
1. Calculate the impedance of the capacitor (Z):
- Impedance (Z) is the effective resistance in an AC circuit and depends on the frequency (w) and the capacitance (C).
- The impedance of a capacitor is given by the formula Z = 1 / (w * C), where w is the angular frequency in rad/s and C is the capacitance in Farads.
- Substituting the given values: C = 1 microFarad and w = 10 rad/s, we get Z = 1 / (10 * 10^(-6)) = 10^5 ohms.
2. Calculate the amplitude of the voltage across the resistor (V0):
- Ohm's Law states that V = I * R, where V is the voltage, I is the current, and R is the resistance.
- In an R-C circuit, the current and voltage are not in phase due to the impedance of the capacitor. However, the amplitude of the current (I0) and voltage (V0) are related.
- Since the resistor and capacitor are connected in series, the amplitude of the current is equal to the amplitude of the input voltage divided by the impedance: I0 = Vin / Z.
- Substitute the given values: Vin = 110 V and Z = 10^5 ohms, we get I0 = 110 / (10^5) = 1.1 * 10^(-3) Amps.
- Since the resistor and capacitor are connected in series, the amplitude of the voltage across the resistor is equal to the amplitude of the current times the resistance: V0 = I0 * R.
- Substitute the given value: R = 1000 ohms, we get V0 = (1.1 * 10^(-3)) * 1000 = 1.1 V.
So, in an R-C circuit driven by an alternating voltage with an amplitude of 110 V and a frequency of 10 rad/s, the amplitude of the voltage across the resistor (V0) is 1.1 V.
Xc = 1/WC = 1/(10*1*10^-6) = = 1*10^5 Ohms.
Z = R-jXc = 1000 - j100,000 = 100,005 Ohms[-89.4o]
I = E/Z = 110/100,005 = 0.0011A
Vo = I*R = 0.0011 * 1000 = 1.1 Volts.