Question

The familiar phenomenon of a rainbow results from the diffraction of sunlight through raindrops. (a) Does the wavelength of light increase or decrease as we proceed outward from the innermost band of the rainbow? (b) Does the frequency of light increase or decrease as we proceed outward? (c) Suppose that instead of sunlight, the visible light from a hydrogen discharge tube (Figure 6.10 ) was used as the light source. What do you think the resulting "hydrogen discharge rainbow" would look like? [Section 6.3\(]\)

Short answer

(a) The wavelength of light increases as we proceed outward from the innermost band of the rainbow. (b) The frequency of light decreases as we proceed outward from the innermost band of the rainbow. (c) The "hydrogen discharge rainbow" would have discrete bands corresponding to the specific wavelengths of light emitted by the hydrogen discharge tube instead of a continuous spectrum of colors.

Step-by-step solution

(a) Wavelength of light in the rainbow

In a rainbow, the innermost band is violet, and the outermost band is red. The wavelength of light increases as we proceed from violet to red, so the wavelength of light increases as we proceed outward from the innermost band of the rainbow.

(b) Frequency of light in the rainbow

The frequency of light is inversely proportional to its wavelength, i.e., \(v = \frac{c}{\lambda}\), where \(v\) is the frequency, \(c\) is the speed of light, and \(\lambda\) is the wavelength. Since the wavelength of light increases as we proceed outward from the innermost band, the frequency of light will decrease as we go from the innermost band (violet) to the outermost band (red).

(c) Hydrogen discharge rainbow

If we were to use visible light from a hydrogen discharge tube as the light source, the resulting rainbow would look different from a typical rainbow formed by sunlight. Sunlight contains a continuous spectrum of colors, while the visible light from a hydrogen discharge tube includes only specific wavelengths corresponding to the Balmer series. The hydrogen discharge rainbow would have discrete bands corresponding to the specific wavelengths of light emitted by the hydrogen discharge tube instead of a continuous spectrum of colors.

Key concepts

Wavelength of Light

When we admire the brilliant spectrum of colors in a rainbow, we're actually observing a natural display of the different wavelengths of light. Wavelength, commonly denoted by the Greek letter \( \lambda \), is defined as the distance between two consecutive crests or troughs in a wave.

Imagine a wave in the ocean; the wavelength would be the distance between the peaks of two consecutive waves. In the context of visible light, wavelength determines color. Short wavelengths correspond to blue and violet light, while long wavelengths correspond to red light. This is why, in a rainbow, the color sequence from inner to outer bands shifts from violet to red. As a result, the wavelength of light increases as we move outward from the innermost band of the rainbow.

The increase in wavelength is a result of the dispersion of light through water droplets in the atmosphere. Each wavelength of light bends at slightly different angles when passing through water, separating white sunlight into its component colors—forming a rainbow. This phenomenon is not just captivating but also illustrates an important principle of light's behavior.

Frequency of Light

The frequency of light, often denoted by \( v \), is the speed at which the crests or troughs of the wave pass by a point. It reflects how quickly the electromagnetic wave oscillates. The speed of light, represented by \( c \), is a fundamental constant, approximately \( 3 \times 10^8 \) meters per second.

According to the relationship \( v = \frac{c}{\lambda} \), the frequency is inversely proportional to the wavelength. This scientific equation means that as the wavelength gets longer (as we move from violet to red in a rainbow), the frequency gets lower, and vice versa. For example, the higher frequency of violet light is what gives it more energy compared to red light. This relationship is crucial in understanding not only rainbows but the entire electromagnetic spectrum. Therefore, as you proceed outward in a rainbow from violet to red, the frequency of light decreases.

While this concept can seem abstract, thinking of it in terms of musical notes can be helpful. If wavelength were pitch, color changes in a rainbow would resemble a scale moving from a high-pitched note (violet) to a lower-pitched note (red).

Hydrogen Discharge Tube

A hydrogen discharge tube is a captivating tool used to explore atomic physics, emitting light as hydrogen gas inside the tube gets excited and then releases energy. When an electric current is passed through the tube, it excites the hydrogen atoms. As these atoms return to their lower energy states, they emit photons—a process known as emission. The emitted photons correspond to specific wavelengths, resulting in a spectral signature unique to hydrogen. This is often examined in high school or college physics labs.

The distinct wavelengths emitted from a hydrogen discharge tube are associated with transitions of electrons between different energy levels in the hydrogen atom. These transitions emit photons at specific wavelengths, known famously as the Balmer series in the visible range. If used to create a rainbow, this light source would produce a 'rainbow' composed of discrete bands, each matching the particular wavelengths of the Balmer series, rather than a continuous spectrum seen in a typical rainbow.

Therefore, a 'hydrogen discharge rainbow' would not exhibit the gradation of colors we associate with a normal rainbow. Instead, it would show distinct lines or bands of color, each corresponding to a particular electron transition in the hydrogen atom, a striking contrast to the continuous blend of colors produced by sunlight.