An AC voltage source of variable angular frequency omega and fixed amplitude V_0 is connected in ser

Q: Why doesn't the resistance of the bulb change with frequency?

Resistance is an inherent property of the material and its dimensions. Unlike reactance, it does not depend on how fast the current is alternating.

Q: What if the frequency was decreased instead?

If the frequency was decreased, the capacitive reactance would increase, leading to a higher total impedance. The current would drop, and the bulb would glow dimmer.

Q: Does the amplitude $V_0$ affect the change in brightness?

The amplitude $V_0$ determines the baseline brightness. However, the *change* in brightness as frequency varies is purely dictated by the changing impedance, regardless of the fixed $V_0$.

Q: Why do we use RMS current for power instead of peak current?

In AC circuits, power fluctuates continuously. Using RMS (Root Mean Square) current gives us the steady equivalent DC current that would dissipate the same average power.

Q: Would an inductor behave the same way?

No! An inductor's reactance is $X_L = \omega L$, which is directly proportional to frequency. Increasing frequency would increase impedance, making the bulb dimmer.

Question

An AC voltage source of variable angular frequency $\omega$ and fixed amplitude $V_0$ is connected in series with a capacitance $C$ and an electric bulb of resistance $R$ (inductance zero). When $\omega$ is increased,

Options:

(A) the bulb glows dimmer

(B) the bulb glows brighter

(C) total impedance of the circuit is unchanged

(D) total impedance of the circuit increases

Sigma Percentile · Accepted Answer

Imagine you are standing in front of a simple electrical setup: an alternating current (AC) source connected in series with a capacitor and a light bulb. The bulb acts as a pure resistor.

The question poses a fascinating scenario: what happens to the brightness of the bulb when we crank up the angular frequency, $\omega$, of the AC source?

To answer this, we need to embark on a logical journey through the principles of AC circuits. Let's break it down step by step.

### Setting the Stage: The AC Circuit

In a direct current (DC) circuit, resistance is the only opposition to current flow. However, in an AC circuit, components like capacitors and inductors introduce an additional form of opposition called **reactance**.

Together, resistance and reactance combine to form the total opposition to current, known as **impedance**, denoted by $Z$.

For our series RC circuit, the impedance is given by the master equation:
$$Z = \sqrt{R^2 + X_C^2}$$

Here, $R$ is the resistance of the bulb, and $X_C$ is the capacitive reactance.

### The Frequency Dance: Capacitive Reactance

The key to unlocking this problem lies in understanding how the capacitor behaves when the frequency changes.

The capacitive reactance $X_C$ is inversely proportional to the angular frequency $\omega$:
$$X_C = \frac{1}{\omega C}$$

This equation tells us a beautiful physical truth: **capacitors resist low frequencies but allow high frequencies to pass more easily**.

When we increase the angular frequency $\omega$, the denominator in our equation grows larger. Consequently, the capacitive reactance $X_C$ decreases.

### The Heart of the Matter: Impedance

Now, let's bring this back to our impedance equation.

The resistance $R$ of the bulb is a fixed physical property; it does not care about the frequency of the AC source.

Since $X_C$ is decreasing while $R$ remains constant, the overall value under the square root becomes smaller.

Therefore, the total impedance $Z$ of the circuit decreases:
$$Z \downarrow$$

The circuit is now offering less total opposition to the flow of alternating current.

### The Final Verdict: Let There Be Light!

With less opposition, what happens to the current?

According to Ohm's law for AC circuits, the RMS current is given by:
$$I_{rms} = \frac{V_{rms}}{Z}$$

Since the impedance $Z$ has decreased, the RMS current $I_{rms}$ flowing through the circuit must increase.

Finally, the brightness of the bulb is determined by the electrical power it dissipates as heat and light. The power dissipated by a resistor is:
$$P = I_{rms}^2 R$$

With a higher current flowing through the same resistance $R$, the power dissipated increases significantly.

**Therefore, the bulb glows brighter!**

This elegant chain of logic—from frequency to reactance, to impedance, to current, and finally to power—perfectly illustrates the dynamic and interconnected nature of alternating current circuits.

Animated Solution for Physics - Electromagnetic Induction: An AC voltage source of variable angular frequency $ω$ and fixed amplitude $V_{0}$ is connected in series with a capacitance $C$ and an electric bulb of resistance $R$ (inductance zero). When $ω$ is increased,

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Visualized Solution

$RC Series Circuit$

$Impedance of RC Circuit$

$Capacitive Reactance$

$Effect of increasing ω$

$Effect on Impedance Z$

$Effect on Current I_{r m s}$

$Power Dissipated$

$What if it was an inductor?$

The Sigma Insight: Alternating Current (AC) and Voltage

Setting the Stage

The Frequency Dance

The Heart of the Matter

The Final Verdict

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List-I

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } RC Series Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Impedance of RC Circuit

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Capacitive Reactance

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Effect of increasing ω

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Effect on Impedance Z

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Effect on Current Irms​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Power Dissipated

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } What if it was an inductor?