Problem 4 - Applied Physics

Question

For an ac circuit with resistance \(145 \mathrm{~m} \Omega\), capacitance \(10.0 \mu \mathrm{F}\), frequency \(72.5 \mathrm{kHz}\), and \(E=7.00 \mathrm{mV}\) : (a) find the impedance (in ohms), (b) find the phase angle, and (c) find the current.

Step-by-Step Solution

Answer

(a) Impedance \(\approx 0.263\, \Omega\); (b) Phase Angle \(\approx -56.31^\circ\); (c) Current \(\approx 26.6\, \mathrm{mA}\).

Step 1: Convert Units

First, convert the given units into standard SI units. The resistance is given as \(145 \mathrm{~m} \Omega\), which equates to \(0.145 \Omega\). The capacitance is given as \(10.0 \mu \mathrm{F}\), which equates to \(10.0 \times 10^{-6} \mathrm{~F}\). The emf \(E\) is \(7.00 \mathrm{mV}\), equating to \(7.00 \times 10^{-3} \mathrm{~V}\).

Step 2: Calculate Capacitive Reactance

The capacitive reactance, \(X_c\), is given by the formula \( X_c = \frac{1}{2\pi f C} \), where \(f\) is the frequency and \(C\) is the capacitance. Substituting the given values, we get:\[X_c = \frac{1}{2\pi \times 72.5 \times 10^3 \times 10.0 \times 10^{-6}}\approx 0.219 \Omega\]

Step 3: Calculate Impedance

The total impedance \(Z\) in an AC circuit with resistance and capacitive reactance is calculated using the formula \( Z=\sqrt{R^2 + X_c^2} \). Substituting the known values:\[Z = \sqrt{(0.145)^2 + (0.219)^2} \approx 0.263 \Omega\]

Step 4: Calculate Phase Angle

The phase angle \(\phi\) is given by the formula \(\phi = \arctan\left(\frac{-X_c}{R}\right)\). Using the calculated values for \(R\) and \(X_c\),\[\phi = \arctan\left(\frac{-0.219}{0.145}\right) \approx -56.31^\circ \]

Step 5: Calculate Current

The current \(I\) in the circuit is given by Ohm's law for AC circuits, \(I = \frac{E}{Z}\). Substituting \(E = 7.00 \times 10^{-3}\) V and \(Z = 0.263 \Omega\) :\[I = \frac{7.00 \times 10^{-3}}{0.263} \approx 0.0266 \text{ A or } 26.6 \text{ mA}\]

Key Concepts

Impedance Calculation

In AC circuits, impedance is a crucial concept that extends the idea of resistance to AC systems. Impedance (Z) is the total opposition that a circuit offers to the flow of alternating current, composed of both resistive (R) and reactive components.
When dealing with circuits containing resistors and capacitors, the impedance can be calculated using the formula:
\[Z = \sqrt{R^2 + X_c^2}\]where \(X_c\) is the capacitive reactance. It combines both the resistive part and the capacitive part of the impedance.

• Resistive Part: Represents the energy dissipated as heat in the resistors.
• Reactive Part: Represents the energy stored and released by the capacitors.

The unit of impedance, like resistance, is Ohms (\(\Omega\)). By knowing the resistance and reactance, you can find the total impedance that tells us how much the circuit resists the AC flow.

Phase Angle

Phase angle (\phi) helps in understanding the relationship between the voltage and current waveforms in AC circuits. It is a measure of how much one waveform leads or lags another waveform.
The phase angle between the voltage and current in a circuit with capacitive reactance is given by:
\[\phi = \arctan\left(\frac{-X_c}{R}\right)\]

• When \(\phi\) is negative, the current leads the voltage (which is typical for capacitive circuits).
• The phase angle provides insight into power factor and efficiency in AC circuits.

Phase angle values range from -90° to 90°, with capacitive and inductive elements drastically shifting the angle below 0° or above, respectively.

Capacitive Reactance

Capacitive reactance (X_c) is the opposition offered by a capacitor in an AC circuit. Unlike resistance, which is constant, capacitive reactance changes with frequency. It is calculated using the formula:
\[X_c = \frac{1}{2\pi f C}\]where \(f\) is the frequency in hertz (Hz) and \(C\) is the capacitance in farads (F).

• As frequency increases, \(X_c\) decreases, meaning the capacitor offers less opposition.
• As capacitance increases, \(X_c\) also decreases.

Capacitive reactance is measured in Ohms (\(\Omega\)). It is crucial in designing and analyzing circuits, especially in filtering and tuning applications.

Ohm's Law in AC Circuits

Ohm's law in AC circuits expands to include the effects of impedance. The fundamental formula is:
\[I = \frac{E}{Z}\]where \(I\) is the current, \(E\) is the electromotive force (EMF) or voltage source, and \(Z\) is the impedance.
This law illustrates the direct relationship between current, voltage, and impedance in AC circuits.

• Increased impedance results in a lower current for a given voltage.
• Similarly, a higher EMF at the same impedance increases the current.

Understanding and applying Ohm's law to any AC circuit help determine how components will behave when subjected to alternating current.