Question

An ac voltage source is connected in series with a 2.0-μF capacitor and a 750-Ω resistor. Using a digital ac voltmeter, the voltage source is measured to be 4.0 V rms, and the voltages across the resistor and across the capacitor are found to be 3.0 V rms and 2.7 V rms, respectively. Determine the frequency of the ac voltage source. Why is the voltage measured across the voltage source not equal to the sum of the voltages measured across the resistor and across the capacitor?

Accepted Answer

Step 1: Understand the problem. The circuit consists of an AC voltage source connected in series with a resistor and a capacitor. The given values are: capacitance $$ C = 2.0 \mu F $$, resistance $$ R = 750 \Omega $$, source voltage $$ V_{source} = 4.0 	ext{ V rms} $$, voltage across the resistor $$ V_R = 3.0 	ext{ V rms} $$, and voltage across the capacitor $$ V_C = 2.7 	ext{ V rms} . $$The goal is to find the frequency $$ f  of $$the AC source and explain why the source voltage is not the arithmetic sum of $$ V_R $$ and $$ V_C . $$Step 2: Recall the relationship between voltages in an AC series circuit. In such a circuit, the voltages across the resistor and capacitor are not simply added algebraically because they are out of phase. The total voltage is the vector sum of $$ V_R $$ and $$ V_C $$, calculated using the Pythagorean theorem: $$ V_{source} = \sqrt{V_R^2 + V_C^2} . $$Verify this relationship using the given values. Step 3: Calculate the capacitive reactance $$ X_C $$, which is given by $$ X_C = \frac{1}{2 \pi f C} . $$Rearrange this formula to solve for the frequency $$ f $$: $$ f = \frac{1}{2 \pi X_C C} . To $$find $$ X_C $$, use Ohm's law for the capacitor: $$ V_C = I X_C $$, where $$ I  is $$the current in the circuit. The current can also be found using $$ V_R = I R $$, so $$ I = \frac{V_R}{R} . $$Substitute $$ I $$ into the equation for $$ X_C . $$Step 4: Substitute the known values into the equations. First, calculate the current $$ I $$ using $$ I = \frac{V_R}{R} . $$Then, use $$ X_C = \frac{V_C}{I}  to $$find the capacitive reactance. Finally, substitute $$ X_C $$ and $$ C $$ into the formula for $$ f  to $$determine the frequency of the AC source. Step 5: Explain why the source voltage is not the arithmetic sum of $$ V_R $$ and $$ V_C . In an AC $$circuit, the resistor's voltage is in phase with the current, while the capacitor's voltage lags the current by 90°. This phase difference means the voltages are not aligned and must be combined as vectors, not scalars. The total voltage is the resultant of these two perpendicular components, calculated using the Pythagorean theorem.

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