Problem 1 - Applied Physics

Question

What is the maximum voltage in an ac circuit in which the instantaneous voltage at \(\theta=35.0^{\circ}\) is \(27.0 \mathrm{~V}\) ?

Step-by-Step Solution

Answer

The maximum voltage \( V_{max} \) is approximately 47.1 V.

Step 1: Identify the Equations

In an AC circuit, the instantaneous voltage can be expressed as \( V = V_{max} \sin \theta \), where \( V_{max} \) is the maximum voltage and \( \theta \) is the phase angle in degrees or radians. Our task is to determine \( V_{max} \).

Step 2: Express the Known Values

We know that at \( \theta = 35.0^{\circ} \), the instantaneous voltage \( V = 27.0 \mathrm{~V} \). Thus, we can substitute these values into the equation to get: \( 27.0 = V_{max} \sin 35.0^{\circ} \).

Step 3: Solve for Maximum Voltage

Rearrange the equation to solve for \( V_{max} \): \( V_{max} = \frac{27.0}{\sin 35.0^{\circ}} \).

Step 4: Calculate \( V_{max} \)

Calculate \( \sin 35.0^{\circ} \) using a calculator, which is approximately \( 0.5736 \). Then, divide 27.0 by this result: \( V_{max} = \frac{27.0}{0.5736} \approx 47.1 \mathrm{~V} \).

Key Concepts

Instantaneous Voltage

In an AC circuit, voltage changes over time, and this changing voltage is what we call "instantaneous voltage." At any given moment, the voltage has a particular value that can be calculated using certain parameters. When we use the term "instantaneous," it just means the voltage at a specific point in time.
This is often expressed through the equation:

• \( V = V_{max} \sin \theta \)

where \( V \) is the instantaneous voltage, \( V_{max} \) is the maximum voltage, and \( \theta \) is the phase angle.
For example, in the given exercise, at a specific angle of 35 degrees, the voltage is 27.0 V. Here, it tells us what the circuit's voltage is at that instant, distinguished from its maximum or average values over time.

Maximum Voltage

Maximum voltage, as the term suggests, is the highest amplitude of voltage that the AC circuit can reach. When electrical engineers or technicians discuss an AC voltage, they often refer to this peak value because it helps gauge the largest potential difference available in the circuit.
To determine this from the instantaneous voltage, we use the sine function equation:

• Given \( 27.0 = V_{max} \sin 35.0^{\circ} \)
• Rearranging, we find \( V_{max} = \frac{27.0}{\sin 35.0^{\circ}} \)

By calculating this, we find the peak or "maximum" voltage the circuit can achieve, which in this case is approximately 47.1 V.
This understanding is crucial: it helps you predict the highest energy level the circuit might need to handle.

Phase Angle

The phase angle, often denoted as \( \theta \), is a measure of how far along its cycle the AC wave is. In simpler terms, it tells us the position within one complete cycle of the wave at any given moment.
Phase angle is critical in conjunction with the trigonometric functions as seen in the equation for instantaneous voltage.

• \( V = V_{max} \sin \theta \)

For example, at \( \theta = 35.0^{\circ} \), the wave is in a specific part of its oscillation. Understanding where you are in the cycle helps with synchronizing and timing circuits that depend on AC voltage fluctuations, such as the timing of signals in a TV broadcast.

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and are fundamental in AC circuit analysis. For our exercise, the sine function \( \sin \theta \) is key. It provides a relationship between the phase angle and the proportion of the maximum voltage reached at any point in time.
Here's how we use it:

• In our equation \( V = V_{max} \sin \theta \), the sine function varies from -1 to 1.
• When it's 1, the instantaneous voltage equals the maximum voltage, \( V = V_{max} \).
• When it's 0, the instantaneous voltage is 0.

Interpreting these results helps us understand how the voltage behaves over time, as it oscillates between its maximum positive and negative values around zero. Knowing this allows for precise calculations in AC circuits, which are essential for designing efficient electrical systems.