A capacitor is connected across an ac source that has voltage amplitude 60.0 V and frequency 80.0 Hz. (b) What is the capacitance C of the capacitor if the current amplitude is 5.30 A?

Verified step by step guidance

1. Identify the given values: voltage amplitude (V) = 60.0 V, frequency (f) = 80.0 Hz, and current amplitude (I) = 5.30 A.
2. Recall the formula for the capacitive reactance, $X_C = \frac{1}{2\pi f C}$, where $f$ is the frequency and $C$ is the capacitance.
3. Use Ohm's Law for AC circuits, which relates the voltage amplitude and current amplitude through the capacitive reactance: $V = I \times X_C$. Rearrange this to solve for $X_C$: $X_C = \frac{V}{I}$.
4. Substitute the values of $V$ and $I$ into the rearranged formula to calculate $X_C$.
5. Finally, solve for the capacitance $C$ using the formula $C = \frac{1}{2\pi f X_C}$. Substitute the values of $f$ and the calculated $X_C$ to find the capacitance.

Key Concepts

Capacitance

Capacitance is the ability of a capacitor to store electrical charge per unit voltage. It is measured in farads (F) and is defined as the ratio of the charge (Q) stored on one plate of the capacitor to the voltage (V) across the plates, expressed as C = Q/V. In AC circuits, capacitance affects how the capacitor responds to changing voltages and currents.

AC Circuit Behavior

In alternating current (AC) circuits, the voltage and current vary sinusoidally with time. The relationship between voltage, current, and impedance in an AC circuit is governed by Ohm's law, modified for capacitors. The current leads the voltage by a phase angle of 90 degrees in a purely capacitive circuit, which is crucial for calculating the current amplitude based on voltage and capacitance.

Impedance in Capacitive Circuits

Impedance (Z) in a capacitive circuit is a measure of how much the circuit resists the flow of current when an AC voltage is applied. For a capacitor, the impedance is given by Z = 1/(ωC), where ω is the angular frequency (ω = 2πf). This relationship allows us to connect the current amplitude (I) and voltage amplitude (V) through the formula I = V/Z, which is essential for solving the given problem.