A resistance of 10 ohms and a capacitance of 0.1 farad are connected in series to
an alternating voltage source with output (in volts) Vt 50 cos30t. Find the current in the
circuit at any time t 0.
To find the current in the circuit at any time t, we will use Ohm's Law and the relationship between current and voltage in a circuit with a resistor and capacitor in series.
First, let's determine the impedance of the circuit since we have both a resistor and a capacitor.
Impedance (Z) is the total opposition to the flow of alternating current (AC) in a circuit. In a circuit with a resistor and capacitor in series, the impedance is given by the formula:
Z = √(R^2 + (1/ωC)^2)
where R is the resistance, C is the capacitance, and ω is the angular frequency of the AC source.
Given:
Resistance (R) = 10 ohms
Capacitance (C) = 0.1 farad
Voltage (V(t)) = 50 cos(30t)
The angular frequency (ω) can be calculated using the formula ω = 2πf, where f is the frequency of the AC source.
Given:
Frequency (f) = 30 Hz
Calculating ω:
ω = 2πf = 2π(30) = 60π rad/s
Now, we can calculate the impedance (Z):
Z = √(R^2 + (1/ωC)^2)
Z = √(10^2 + (1/(60π*0.1))^2)
Z = √(100 + (1/(6π))^2)
Z ≈ √(100 + 0.017)^2
Z ≈ √(100 + 0.000289)
Z ≈ √100.000289
Z ≈ 10
The impedance of the circuit is approximately 10 ohms.
Now, we can find the current (I) using Ohm's Law:
I = V(t) / Z
Substituting the given AC voltage into the equation:
I = (50 cos(30t)) / 10
I = 5 cos(30t)
Therefore, the current in the circuit at any time t is given by I = 5 cos(30t).