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).