Problem 41 - Physics

Question

A \(40.0-\mu F\) capacitor is connected across a 60.0 -Hz generator. An inductor is then connected in parallel with the capacitor. What is the value of the inductance if the rms currents in the inductor and capacitor are equal?

Step-by-Step Solution

Answer

The inductance is approximately 176 mH.

Step 1: Determine the Capacitive Reactance

The capacitive reactance \( X_C \) is given by the formula \( X_C = \frac{1}{2\pi f C} \). Here, \( f = 60.0 \) Hz and \( C = 40.0 \times 10^{-6} \) F. Substituting these values, we get:\[ X_C = \frac{1}{2\pi \times 60.0 \times 40.0 \times 10^{-6}} \approx 66.3 \text{ ohms.} \]

Step 2: Apply Condition for Equal Currents

For the rms currents in the capacitor and inductor to be equal, their reactances must be equal in magnitude. Thus, \( X_L = X_C \), where \( X_L = 2\pi f L \). So, we have:\[ 2\pi \times 60.0 \times L = 66.3. \]

Step 3: Solve for Inductance

Rearranging the equation from Step 2 to find \( L \), we get:\[ L = \frac{66.3}{2\pi \times 60.0}. \] Calculating this gives:\[ L \approx 0.176 \text{ H} \text{ or } 176 \text{ mH}. \]

Key Concepts

Capacitive Reactance

In an AC circuit, capacitive reactance plays a vital role by resisting the change of voltage across a capacitor. It is symbolized by \( X_C \) and essentially hinders the flow of alternating current that charges and discharges the capacitor. The formula for capacitive reactance is \( X_C = \frac{1}{2\pi f C} \), where \( f \) is the frequency in hertz, and \( C \) is the capacitance in farads.

An important thing to note is that capacitive reactance is frequency dependent. This means that as the frequency of the generator increases, the capacitive reactance decreases, allowing more current to pass through. Conversely, at lower frequencies, the reactance is higher, and less current is allowed to pass. The capacitive reactance is measured in ohms, just like resistance, but it is not a real resistance as it does not dissipate energy but instead stores it temporarily.

Inductive Reactance

Inductive reactance, denoted by \( X_L \), is the opposition that an inductor presents to changes in current in an AC circuit. Unlike capacitors that resist changes in voltage, inductors resist changes in current flow. The formula for inductive reactance is \( X_L = 2\pi f L \), where \( f \) is the frequency and \( L \) is the inductance in henrys.

Similar to capacitive reactance, inductive reactance is also frequency dependent. As the frequency increases, the inductive reactance increases, opposing the change in current. At lower frequencies, the reactance is lower, allowing more current to flow. This behavior is because inductors store energy in a magnetic field when current flows through them, and this field resists any changes in the current flow. Inductive reactance is also measured in ohms.

RMS Current

RMS, or root mean square, current is a way of representing the effective value of an alternating current. It is particularly useful because it allows us to compare AC circuits to DC circuits by providing a measure of how much AC voltage is as effective as a certain DC voltage. Mathematically, it is the square root of the average of the squares of instantaneous values over a cycle.

The RMS current in a circuit with capacitive and inductive components is influenced by both the resistive and reactive elements. In the context of the given exercise, it becomes crucial to equate the currents through the inductor and capacitor by matching their respective reactances—ensuring that the effects of the inductive and capacitive components balance each other out.

Parallel Circuit Analysis

When components are connected in parallel in a circuit, they share the same voltage across them but can carry different currents. This characteristic makes parallel circuits quite distinct from series circuits, where the current is the same but the voltage varies across components.

In alternating current circuits, analyzing a parallel circuit involves evaluating the reactance of each component. For the circuit in the exercise, setting the reactances equal ensures that the RMS currents through both the capacitor and inductor are identical.

This method of analysis helps in designing circuits where certain current distributions or phase conditions are desired, using the principles of AC impedance to achieve the specific electrical behavior needed in each branch of the circuit.