For an RC circuit, find an expression for the angular frequency at which V_R = ½ ε₀. What is V_C at this frequency?

Verified step by step guidance

1. Identify the context of the problem: 'VC' likely refers to the voltage across a capacitor in an AC circuit. To solve for VC, we need to understand the circuit configuration and the given frequency.
2. Recall the formula for the capacitive reactance (Xc), which is given by: $Xc=\frac{1}{\left(2\pi fC\right)}$, where f is the frequency and C is the capacitance.
3. Determine the total impedance (Z) of the circuit if it is a series circuit. For a series R-C circuit, the impedance is given by: $Z=\sqrt{R^2+X^c}$, where R is the resistance and Xc is the capacitive reactance.
4. Use Ohm's Law to find the current (I) in the circuit: $I=\frac{V}{Z}$, where V is the total voltage supplied to the circuit.
5. Finally, calculate the voltage across the capacitor (VC) using the formula: $VC=IXc$. Substitute the values of I and Xc to find VC.

Key Concepts

Voltage (V)

Voltage, or electric potential difference, is the measure of electric potential energy per unit charge between two points in an electric field. It is expressed in volts (V) and indicates how much work is needed to move a charge from one point to another. Understanding voltage is crucial for analyzing circuits and determining how electrical components interact at a given frequency.

Capacitance (C)

Capacitance is the ability of a system to store an electric charge, defined as the ratio of the electric charge (Q) stored to the voltage (V) across the system. It is measured in farads (F). In the context of AC circuits, capacitance affects how voltage and current behave at different frequencies, influencing the overall impedance of the circuit.

Frequency (f)

Frequency refers to the number of cycles of a periodic wave that occur in a unit of time, typically measured in hertz (Hz). In electrical circuits, frequency plays a significant role in determining how components like capacitors and inductors respond to alternating current (AC). The relationship between frequency and voltage is essential for understanding the behavior of reactive components in AC circuits.