Intensity of a Wave (7.1.6) | CIE A-Level Physics Notes

Mathematical Definition and Units of Intensity

Intensity, in the context of waves, signifies the amount of energy a wave carries across a unit area perpendicular to the wave's direction of propagation.

Intensity of a wave

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Intensity Formula: Intensity (I) is defined as the power (P) transmitted per unit area (A). The formula I = P/A succinctly captures this relationship.
Understanding Power: Power in this context refers to the rate at which energy is transferred by the wave. It's an essential component in calculating intensity.
Units of Measurement: Intensity is measured in watts per square metre (W/m^2) in the International System of Units (SI). This unit helps quantify the energy transfer rate in a way that is tangible and measurable.

Deep Dive into the Intensity-Amplitude Relationship

The relationship between intensity and amplitude is one of the fundamental concepts in wave physics. It's crucial to understand how changes in a wave's characteristics affect its energy transmission.

Square Dependence: A pivotal aspect of this relationship is that intensity varies as the square of the wave's amplitude. This means that a small increase in amplitude leads to a significantly larger increase in intensity.
Mathematical Representation: For a wave with amplitude (A), the intensity (I) can be represented as I proportional to A squared. This quadratic relationship is central to understanding how energy is distributed in wave phenomena.
Physical Interpretation: The square dependency implies that even a moderate increase in the amplitude of a wave (like sound or light) can result in a substantial increase in its intensity, significantly altering its impact on surroundings.

Real-World Applications and Examples

To contextualise the theoretical aspects, we examine how intensity plays a role in everyday phenomena, particularly focusing on sound and light.

Sound Levels

Intensity and Loudness: In acoustics, the intensity of sound waves is directly linked to their loudness. A higher intensity translates to a louder sound.

Two sound waves with different intensities (low and high)

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Decibels - A Practical Measure: Sound intensity is often measured in decibels (dB), a logarithmic unit. This scale helps in representing the vast range of sound intensities humans can hear.
Example in Daily Life: A whisper, measuring around 30 dB, has much lower intensity compared to a jet engine's roar, which can exceed 120 dB. This demonstrates the practical implications of intensity in our auditory experiences.

Light Brightness

Role in Optics: In optics, the intensity of light waves determines the brightness of a light source.
Inverse-Square Law: The intensity of light diminishes with the square of the distance from its source, a principle known as the inverse-square law. This law is crucial in understanding phenomena like the apparent brightness of stars.
Practical Application: Photographers often adjust their settings based on the intensity of light to achieve the desired exposure in their images. This is a direct application of understanding light intensity.

Intensity in Different Wave Types

Different types of waves, such as mechanical waves (sound) and electromagnetic waves (light), exhibit unique characteristics in how they transfer energy through intensity.

Mechanical Waves

Medium Dependency: For mechanical waves, the intensity also depends on the medium through which the wave is travelling. The density and elasticity of the medium play a significant role in determining how energy is transmitted.
Sound in Various Media: For instance, sound waves travel differently through air, water, and solids, leading to variations in intensity and perceived loudness.

Electromagnetic Waves

Medium Independence: Unlike mechanical waves, electromagnetic waves do not require a medium. Their intensity depends on the amplitude of the electric and magnetic fields.
Solar Radiation: An example of this is the intensity of sunlight, which varies with time of day and atmospheric conditions but remains constant in the vacuum of space.

Intensity in Wave Interference

Constructive and Destructive Interference: When two waves interfere, their intensities can add up (constructive interference) or cancel out (destructive interference), leading to patterns of varying intensity.
Applications in Technology: This principle is used in technologies like noise-cancelling headphones, where destructive interference is used to reduce unwanted sound.

FAQ

The intensity of a wave cannot be negative. Intensity, by definition, is a measure of the energy transfer per unit area and is always a positive value. It represents the amount of energy a wave carries across a given area, and energy transfer cannot be negative. In physics, negative energy transfer is not a concept that aligns with the laws of thermodynamics or wave mechanics. While the amplitude or displacement in a wave can have negative values, indicating direction or phase, intensity, being a squared quantity (proportional to amplitude squared), will always be non-negative.

The phase difference between waves is crucial when considering the intensity resulting from wave interference. When two waves meet, their phase difference determines whether they will undergo constructive or destructive interference. Constructive interference occurs when waves are in phase (phase difference is zero or a multiple of 2π), leading to increased amplitude and, consequently, greater intensity. Destructive interference happens when the waves are out of phase (phase difference is an odd multiple of π), resulting in reduced amplitude and lower intensity. The phase difference, therefore, plays a vital role in applications like noise-cancelling headphones, where destructive interference is used to reduce sound intensity.

The wavelength of a wave is indirectly related to its intensity. While intensity itself is directly proportional to the square of the wave's amplitude, wavelength comes into play when considering the wave's frequency and speed. For a given wave speed, a shorter wavelength implies a higher frequency. Since intensity is also dependent on frequency (for certain types of waves like electromagnetic radiation), a higher frequency can lead to greater intensity. However, this relationship is more nuanced and depends on the type of wave and its specific properties. For example, in electromagnetic waves, intensity can be influenced by both the electric and magnetic field amplitudes, which are related to wavelength and frequency.

Yes, it is possible for two waves to have the same intensity but different amplitudes, particularly when other factors influencing intensity vary. Intensity is proportional to the square of the amplitude, but it also depends on other variables such as frequency and the medium's properties. For instance, in electromagnetic waves, intensity depends on the amplitude of the electric and magnetic fields. Two electromagnetic waves could have different electric and magnetic field amplitudes but still result in the same intensity if the product of these field amplitudes is the same. Similarly, two sound waves might have different amplitudes but the same intensity if they travel in media with different densities or elastic properties.

The medium through which a wave travels significantly influences its intensity. For mechanical waves like sound, the medium's properties, such as density and elasticity, directly affect how energy is transmitted. In denser media, particles are closer together, allowing energy to be transferred more efficiently, potentially increasing the intensity of the wave. Conversely, in less dense media, the energy transfer is less efficient, leading to a decrease in intensity. The medium's elasticity also plays a role; more elastic media can store and transmit energy more effectively. This principle is not applicable to electromagnetic waves, such as light, which do not require a medium and whose intensity is independent of the surrounding environment.

Practice Questions

A speaker emits sound with an intensity of 0.5 W/m^2 at a distance of 2 meters. If the distance from the speaker is doubled, what is the new intensity of the sound? Explain the concept applied.

The new intensity of the sound would be 0.125 W/m². This scenario exemplifies the inverse-square law, which states that the intensity of a wave (like sound) decreases with the square of the distance from its source. When the distance is doubled (from 2 meters to 4 meters), the intensity becomes one-fourth of its original value because the square of 2 (the factor by which the distance is increased) is 4. Hence, the original intensity of 0.5 W/m²is divided by 4, resulting in 0.125 W/m².

A light bulb emits light uniformly in all directions with a total power of 100 watts. Calculate the intensity of light at a distance of 5 meters from the bulb. Briefly describe the process used.

The intensity of the light at a distance of 5 meters from the bulb is 1.27 W/m². To calculate this, use the formula for intensity, I = P/A, where P is the power and A is the area through which the power is spread. As the light spreads uniformly in all directions, the area is the surface area of a sphere, A = 4πr². Substituting the values, A = 4π(5²) = 314.16 m². Thus, I = 100 W / 314.16 m² = 1.27 W/m². This calculation is based on the understanding of how power disperses over an area and the geometric interpretation of the spreading of light.

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