Problem 30 - University Physics with Modern Physics

Question

An electromagnetic standing wave in air of frequency 750 MHz is set up between two conducting planes 80.0 \(\mathrm{cm}\) apart. At which positions between the planes could a point charge be placed at rest so that it would remain at rest? Explain.

Step-by-Step Solution

Answer

Place the charge at 0, 20, 40, 60, or 80 cm between the plates.

Step 1: Understanding Standing Waves

In a standing wave, the positions where the displacement is always zero are called nodes and the positions of maximum displacement are called antinodes. Points at nodes have zero electric field, so a charge placed there will not experience any force.

Step 2: Determine Wavelength

The standing wave is for a frequency of 750 MHz. The speed of light in air is approximately 300,000 km/s. To find the wavelength, use the formula \( \lambda = \frac{c}{f} \), where \( c = 300,000,000 \) m/s and \( f = 750,000,000 \) Hz. Thus, \( \lambda = \frac{300,000,000}{750,000,000} = 0.4 \) m or 40 cm.

Step 3: Node Position Calculation

Nodes in a standing wave occur at integer multiples of \( \frac{\lambda}{2} \). Since the wavelength is 40 cm, nodes occur at: \( 0, 20, 40, 60, 80 \) cm within the 80 cm range between the plates.

Step 4: Conclusion of Placement

A point charge can remain at rest if placed at a node because there is no net electric force at a node position. Thus, a charge can be placed at \( 0, 20, 40, 60, \text{ and } 80 \) cm from one of the plates.

Key Concepts

Standing Waves

Standing waves are a fascinating concept in the world of physics, particularly when dealing with electromagnetic waves. They occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a wave pattern that appears to be stationary, hence the name "standing wave." In this scenario, certain points along the wave, called **nodes**, remain stationary and do not move. The other points, called **antinodes**, experience maximum movement. Understanding where nodes and antinodes occur in standing waves is crucial because around nodes there's no displacement, particularly significant to placing a point charge so that it remains at rest.

Frequency Calculation

Frequency calculation is essential in determining various characteristics of waves, such as wavelength. The frequency (\( f \)) is generally given in Hertz (Hz) and represents the number of cycles a wave completes per second. In the context of our electromagnetic wave, a frequency of 750 MHz corresponds to 750 million cycles per second. Such high frequencies are typical of electromagnetic waves like radio and microwave radiation. Calculating frequency involves knowing the energy and type of wave, but if you're given frequency like in this exercise, it's a stepping stone to finding out more about other properties like wavelength through the speed of light.

Nodes and Antinodes

Nodes and antinodes are specific points along a standing wave. Understanding these helps one grasp how standing waves behave. **Nodes** are points in the wave where the amplitude is constantly zero. At these points, the interference between the two opposing waves results in cancellation. **Antinodes**, on the other hand, are points where the amplitude reaches a maximum. They occur midway between nodes. For a point charge in an electromagnetic wave, placing it at a node means it will remain unaffected by the electric field, while an antinode would lead to maximum motion. In this exercise, nodes were particularly important because they are where the electric field strength is zero, allowing a charge to stay still.

Wavelength Determination

Wavelength is a fundamental property of waves that measures the distance between two consecutive points that are in phase, such as peak-to-peak or trough-to-trough. To determine the wavelength (\( \lambda \)) of a wave, you can use the formula \( \lambda = \frac{c}{f} \), where \( c \) is the speed of light (approximately 300,000,000 m/s in air) and \( f \) is the frequency. For our 750 MHz electromagnetic wave, the wavelength was calculated to be 40 cm. This means that every 40 cm, a full cycle of the wave is completed. Knowing the wavelength helps in calculating the positions of nodes, specifically at intervals of half the wavelength (\( \frac{\lambda}{2} \)), which directly affects the positioning of point charges in the wave field without experiencing force.