Problem 34 - Fundamentals of Physics

Question

A sodium light source moves in a horizontal circle at a constant speed of \(0.100 c\) while emitting light at the proper wavelength of \(\lambda_{0}=589.00 \mathrm{~nm}\). Wavelength \(\lambda\) is measured for that light by a detector fixed at the center of the circle. What is the wavelength shift \(\lambda-\lambda_{0}\) ?

Step-by-Step Solution

Answer

The wavelength shift is \(-2.445 \mathrm{~nm}\).

Step 1: Express the relationship

When an object emitting light moves with speed comparable to the speed of light, we use the relativistic Doppler effect to find the observed wavelength. The formula for the relativistic Doppler shift when the source is moving with velocity (v) perpendicular to the line of sight is \( \lambda = \lambda_{0} \sqrt{1 - \frac{v^2}{c^2}} \) where \( c \) is the speed of light.

Step 2: Substitute the values

Given that \( v = 0.100c \) and \( \lambda_0 = 589.00 \mathrm{~nm} \), substitute into the Doppler shift formula: \( \lambda = 589.00 \mathrm{~nm} \sqrt{1 - (0.100)^2} \).

Step 3: Calculate the factor

Compute \( 1 - (0.100)^2 = 1 - 0.010 = 0.990 \). Then, calculate \( \sqrt{0.990} \).

Step 4: Find the observed wavelength

Substitute the factor \( \sqrt{0.990} \approx 0.995 \) into the wavelength equation: \( \lambda = 589.00 \mathrm{~nm} \times 0.995 \).

Step 5: Compute the observed wavelength

Calculate the product: \( 589.00 \times 0.995 = 586.555 \mathrm{~nm} \).

Step 6: Determine the wavelength shift

The wavelength shift is given by \( \lambda - \lambda_0 = 586.555 - 589.00 \mathrm{~nm} \).

Step 7: Final calculation of the wavelength shift

Subtract the original wavelength from the observed one: \( 586.555 - 589.00 = -2.445 \mathrm{~nm} \).

Key Concepts

Wavelength Shift

Wavelength shift is a key concept in understanding the relativistic Doppler effect. It refers to the change in wavelength observed when a light source moves relative to an observer. In the presence of a moving source, the wavelength of light appears different than when measured from a stationary point. This shift is calculated as the difference between the observed wavelength and the proper wavelength.

In our example, the shift is negative, indicating a blueshift due to the source moving in a circular path. For this particular problem, the observed wavelength was found using the formula for relativistic Doppler shift: \[ \lambda = \lambda_{0} \sqrt{1 - \frac{v^2}{c^2}} \] where \( v \) is the velocity of the source and \( c \) is the speed of light. By implementing this formula, we determine how relative motion affects the perceived color of the light.

Speed of Light

The speed of light, denoted as \( c \), plays a crucial role in modern physics. It is the maximum speed at which all energy, matter, and information in the universe can travel. Measured as approximately \( 299,792,458 \) meters per second (m/s), or about \( 300,000 \) km/s, this constant is fundamental in various equations governing relativity.

In the context of the relativistic Doppler effect, the speed of light is essential for calculating wavelength shifts. It acts as a comparison for the velocity of the moving light source. When the source moves at speeds comparable to \( c \), time dilation and length contraction become significant, affecting the perceived wavelength. This is why understanding the speed of light is necessary to fully grasp the Doppler shift phenomenon. Its invariance also underscores its role as a consistent measure across different frames of reference, ensuring the equations used are universally applicable.

Proper Wavelength

Proper wavelength refers to the wavelength measured in a frame where the light source is at rest. It is the reference point for observing any changes or shifts due to relative movement.

In our exercise, the proper wavelength is given as \( \lambda_{0}=589.00 \mathrm{~nm} \). This is the wavelength of the sodium light in the absence of any motion. When the source moves, specifically at speeds significant relative to the speed of light, the observed wavelength differs. This change is captured by the formula for the relativistic Doppler shift, highlighting how movement alters perception.

The proper wavelength remains a constant, unshifted value from which the observed data can be contrasted. This helps in determining how much the wavelength is affected under different conditions, such as varying speeds or directions of movement. Understanding the proper wavelength ensures accurate analysis of the wavelength shift, as it serves as the baseline measurement in comparison to the shifted wavelength.