Problem 9 - Applied Physics

Question

Find the current (in amperes) in each inductive circuit. \(L=5.00 \mathrm{mH}, f=2.00 \mathrm{kHz}, E=50.0 \mathrm{~V}\)

Step-by-Step Solution

Answer

The current in the circuit is approximately 0.796 A.

Step 1: Calculate Angular Frequency

First, calculate the angular frequency \[ \omega = 2\pi f \]Substituting the given frequency:\[ \omega = 2 \times \pi \times 2000 \]\[ \omega = 4000\pi \] radians per second.

Step 2: Calculate Inductive Reactance

Next, find the inductive reactance using the formula:\[ X_L = \omega L \]Substitute the known values:\[ X_L = 4000\pi \times 5 \times 10^{-3} \]\[ X_L = 20\pi \] ohms.

Step 3: Find Current Using Ohm's Law for AC Circuits

Now, use Ohm's Law for AC circuits to find the current:\[ I = \frac{E}{X_L} \]Substitute the values:\[ I = \frac{50}{20\pi} \]\[ I \approx \frac{50}{62.83} \] \[ I \approx 0.796 \] amperes.

Key Concepts

Inductive Reactance

Inductive reactance is a crucial concept in understanding how inductors behave in AC circuits. It is the property of an inductor that resists the change in current through it. Reactance, represented by the symbol \( X_L \), is measured in ohms, similar to resistance, but it behaves differently.
In an AC circuit, current continuously changes direction and magnitude. Inductive reactance increases with frequency, since the inductor opposes rapid changes in current more strongly than slower ones. The reactance is calculated with the formula:

• \( X_L = \omega L \)

Here, \( \omega \) is the angular frequency, and \( L \) is the inductance of the coil in henrys. In our exercise, increasing either the frequency \( f \) or the inductance \( L \) would increase \( X_L \), affecting the current flow in the circuit differently from straight DC resistance.

Ohm's Law

Ohm's Law is a fundamental principle used to relate voltage, current, and resistance in electrical circuits. In the context of AC circuits involving inductors, the law is slightly modified to accommodate reactance instead of resistance.
This modified form of Ohm's Law is expressed as:

• \( I = \frac{E}{X_L} \)

Where \( I \) is the current in amperes, \( E \) is the voltage across the inductor in volts, and \( X_L \) is the inductive reactance in ohms. It's important to understand that this form applies specifically to AC circuits, where inductive reactance plays a significant role. This adjustment is crucial because it accounts for the opposition due to the inductor's reactance, not just resistive losses as in DC circuits.

Angular Frequency

Angular frequency is a concept used in describing oscillations, such as those occurring in AC circuits. It is often symbolized by \( \omega \) and has the unit of radians per second.
It is calculated from the frequency \( f \) of the AC source using the following formula:

• \( \omega = 2 \pi f \)

This parameter is fundamentally important because it connects the regular frequency with the circular motion idea, which models AC behavior. By incorporating \( 2\pi \), angular frequency translates cycles per second into radians per second, capturing the rotational aspect of alternating currents. A larger \( \omega \) value means faster oscillation, which increases inductive reactance and influences current flow in the circuit.

Inductors

Inductors are passive electrical components that store energy in a magnetic field when electrical current flows through them. They are characterized by their inductance \( L \), which is measured in henrys.
Inductors are commonly used in various electronic devices and AC circuits. Their behavior in AC circuits is defined by their ability to resist changes in current, a phenomenon termed inductive reactance. Key aspects of inductors in AC circuits include:

• Storing energy as a magnetic field when current flows.
• Introducing inductive reactance, which increases with frequency.
• Shifting the phase between voltage and current, causing the current to lag behind the voltage by 90 degrees.

Inductors play a transformative role in circuit dynamics, shaping how devices function by influencing current impedance and energy storage.