Question

A resistor and a capacitor are connected in series with an ac source. If the potential drop across the capacitor is \(5 \mathrm{~V}\) and that across resistor is \(12 \mathrm{~V}\), the applied voltage is, (a) \(13 \mathrm{~V}\) (b) \(17 \mathrm{~V}\) (c) \(5 \mathrm{~V}\) (d) \(12 \mathrm{~V}\)

Short answer

The applied voltage in the given AC circuit can be found using the phasor addition technique and the Pythagorean theorem. Given the potential drop across the capacitor as \(5\,\text{V}\) and across the resistor as \(12\,\text{V}\), we can calculate the applied voltage as \(V_S = \sqrt{(12\,\text{V})^2 + (5\,\text{V})^2} = 13\,\text{V}\). Therefore, the correct answer is (a) \(13\,\text{V}\).

Step-by-step solution

1. Recall the Phasor Representation of Voltages in AC Circuits

In an AC circuit, the voltages across resistive and capacitive elements have different phase relationships to the source voltage, which also varies sinusoidly with time. In a phasor diagram, the voltage across the resistor (V_R) is in the same phase with the source voltage (V_S) while the voltage across the capacitor (V_C) has a phase difference of 90 degrees (pi/2) leading.

2. Use the Phasor Addition to Find the Applied Voltage

Since the voltage across the resistor and capacitor have different phase relationships, we need to use the phasor addition technique to add the voltages vectorially. The phasor diagram forms a right-angled triangle, with V_R being the adjacent side, V_C being the opposite side and V_S (the applied voltage) being the hypotenuse. So, by using the Pythagorean theorem, we can find the applied voltage: V_S = \(\sqrt{(V_R)^2 + (V_C)^2}\)

3. Substitute the Given Values

Substitute the given potential drops across the resistor and capacitor into the equation: V_S = \(\sqrt{(12\,\text{V})^2 + (5\,\text{V})^2}\)

4. Calculate the Applied Voltage

Solve the equation to calculate the applied voltage: V_S = \(\sqrt{(144\,\text{V}^2) + (25\,\text{V}^2)}\) = \(\sqrt{169\,\text{V}^2}\) = 13 V Hence, the applied voltage is 13 V, which corresponds to the option (a).

Key concepts

Phasor Diagram

A phasor diagram is a visual representation of the phase relationships between different voltages or currents in an AC circuit. Imagine it like a snapshot capturing how alternating currents or voltages change with time.

• The voltages are shown as vectors called phasors, which rotate with a constant angular velocity.
• This tool allows you to see how resistive and reactive components like capacitors interact in an AC circuit.

In the phasor diagram, the voltage across a resistor is always in phase with the current and source voltage. This is because Ohm's law says both the current and voltage change in the same way over time.
For a capacitor, however, the voltage lags the current by a phase difference of 90 degrees or \(\pi/2\) radians. This means the current wave reaches its peak a quarter cycle before the voltage does. When these voltages are drawn on a phasor diagram, the line representing the capacitor voltage is perpendicular to the line representing the resistor voltage.
Understanding these phase differences is important as it helps in calculating the total or applied voltage in circuits where multiple components are connected in series.

Resistor-Capacitor Series Circuit

In an AC circuit where a resistor and capacitor are connected in series, they share the same current but have different voltage drops and phase relationships.

• The resistor part of the circuit has its voltage drop in phase with the current and the source voltage.
• The capacitor part has its voltage drop lagging behind the current by 90 degrees or \(\pi/2\) radians.

This series combination can be analyzed using phasors. Because the voltage variations across these two components are not in phase, their resultant or applied voltage cannot be found simply by arithmetic addition.
Instead, we have to perform a phasor addition, taking into account their phase relationship, which is typically achieved using vector addition. The diagram formed by these phasors often creates a right-angled triangle, which is crucial for determining the combined voltage using trigonometric methods.

Pythagorean Theorem in AC Analysis

When analyzing AC circuits, especially those involving resistors and capacitors, the Pythagorean theorem becomes a valuable tool. This is because when voltage or current phasors form a right-angled triangle, this theorem helps find the unknown quantities.

• In the phasor diagram of a resistor-capacitor series circuit, this theorem aids in calculating the total voltage across both components.
• The resistor's voltage and the capacitor's voltage are the two shorter sides of the right triangle, corresponding to the adjacent and opposite sides respectively.

In our scenario, the voltages across the resistor and capacitor are respectively given as \(12 \, \text{V}\) and \(5 \, \text{V}\). The Pythagorean theorem, which states that the square of the hypotenuse (i.e., total voltage \(V_S\)) is equal to the sum of the squares of the other two sides, can be used here:
\[ V_S = \sqrt{(V_R)^2 + (V_C)^2} \]
Plugging in our values:
\[ V_S = \sqrt{(12 \, \text{V})^2 + (5 \, \text{V})^2} = 13 \, \text{V} \]
This equation shows how phasor relationships and the Pythagorean theorem are powerful in determining the resulting effect of connecting passive elements in AC circuits.