Question

Show that the power dissipated in a resistor connected to an AC power source with frequency \(\omega\) oscillates with frequency \(2 \omega\).

Short answer

Answer: The frequency of power dissipation in a resistor connected to an AC power source with frequency ω is 2ω.

Step-by-step solution

1. Voltage across the resistor

The voltage across the resistor connected to an AC power source can be represented as: \(v(t) = V_m \cos(\omega t)\) where \(V_m\) is the maximum voltage and \(\omega\) is the angular frequency of the source.

2. Current through the resistor

According to Ohm's Law, the current through the resistor can be found by dividing the voltage across the resistor by its resistance: \(i(t) = \frac{v(t)}{R} = \frac{V_m \cos(\omega t)}{R}\) where \(R\) is the resistance of the resistor.

3. Power dissipated in the resistor

The power dissipated in the resistor can be found using the formula: \(P(t) = v(t) \cdot i(t) = \frac{V^2_m}{R}\cos^2(\omega t)\)

4. Expressing power in terms of trigonometric identity

We can simplify the expression further by using the double-angle trigonometric identity: \(\cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2}\) So, substituting the trigonometric identity into the power equation, we get: \(P(t) = \frac{V^2_m}{R}\left (\frac{1 + \cos(2\omega t)}{2}\right)\)

5. Frequency of power dissipation

Now, we can see that the power dissipation function has a term \(\cos(2\omega t)\), which means that the power dissipated in the resistor oscillates with a frequency of \(2\omega\). So, we have shown that the power dissipated in a resistor connected to an AC power source with frequency \(\omega\) oscillates with frequency \(2\omega\).

Key concepts

Oscillating Power

In an AC (Alternating Current) circuit, the power dissipated in a resistor is not constant over time. Instead, it varies in a specific pattern, often referred to as oscillating power. This oscillation is due to the alternating nature of the voltage and current in AC circuits.

When analyzing how power behaves, we use the equation for instantaneous power, which in this case becomes dependent on both the voltage and current going through the resistor:

• Voltage over time, given by: \(v(t) = V_m \cos(\omega t) \)
• Current over time, following Ohm's Law, becomes: \(i(t) = \frac{V_m \cos(\omega t)}{R} \)

Plugging these into the power equation \(P(t) = v(t) \cdot i(t)\), we derive that power not only depends on time but involves trigonometric identities to fully describe its behavior.

This results in a power expression with components oscillating at twice the frequency of the original voltage and current source. Hence, if the original frequency is \(\omega\), the power oscillates at \(2\omega\). The variation demonstrates how energy gets periodically absorbed and released, which is a characteristic feature of AC system dynamics.

Ohm's Law

Ohm's Law is fundamental to electrical circuits, serving as the primary connection between voltage, current, and resistance. It states that the current \(i\) flowing through a conductor between two points is directly proportional to the voltage \(v\) across the two points and inversely proportional to the resistance \(R\) of the conductor:

\( i(t) = \frac{v(t)}{R} \).

This formula is crucial when analyzing AC circuits as it allows us to determine the instantaneous current given an AC voltage.

In our AC scenario, the application of Ohm's Law transforms the time-varying voltage \(v(t) = V_m \cos(\omega t)\) to a time-varying current:

• The expression for the current: \(i(t) = \frac{V_m \cos(\omega t)}{R}\).

With this law, alongside other equations, it becomes possible to compute the instantaneous power in the resistor, bridging the basic concept of resistance to more complex AC circuit analyses.
Ohm's Law in AC circuits reflects how resistors resist the flow of alternating current, affecting how power is dissipated over time.

Trigonometric Identities

Trigonometric identities are mathematical tools that help simplify the computation of AC circuit elements, such as power. One such identity is the double-angle formula, which was crucial in demonstrating the oscillation frequency of power in this exercise.

The key identity used is:
\( \cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2} \).
This is known as the power-reduction formula and can transform expressions of squared trigonometric functions into expressions involving angles of double frequency.

• In the exercise, substituting \(\cos^2(\omega t)\) in the power equation with its trigonometric identity enabled us to see that the term involving \(\cos(2\omega t)\) dictates that the power oscillates at twice the frequency of the source \(\omega\).
• This substitution provides a much simpler form that highlights how the components of power vary over time.

By using these identities, complex AC power equations become more manageable, enhancing our understanding of how different factors within the circuit relate and influence each other over time.