Problem 2 - University Physics with Modern Physics

Question

In a TV picture, ghost images are formed when the signal from the transmitter travels to the receiver both directly and indirectly after reflection from a building or other large metallic mass. In a 25 -inch set, the ghost is about 1.0 \(\mathrm{cm}\) to the right of the principal image if the reflected signal arrives 0.60\(\mu\) s after the principal signal. In this case, what is the difference in path lengths for the two signals?

Step-by-Step Solution

Answer

The difference in path lengths is 180 meters.

Step 1: Understanding the Problem

We're asked to find the difference in path lengths between a direct signal and a reflected signal that results in a ghost image on a TV screen. The reflected signal arrives 0.60 microseconds later than the direct signal.

Step 2: Recognize the Given Values

The time delay between the direct and reflected signals is 0.60 microseconds (\(0.60 \times 10^{-6}\) seconds). We will also use the speed of light since these signals travel at this speed: \(c = 3.00 \times 10^8\) m/s.

Step 3: Calculate the Distance Travelled in Delay Time

Since the radio signal travels at the speed of light, we use the formula \distance = speed \times time\. Thus: \(\Delta d = c \times \Delta t\). We substitute in the known values: \(\Delta d = 3.00 \times 10^8\, \text{m/s} \times 0.60 \times 10^{-6}\, \text{s}\).

Step 4: Compute Path Difference

Calculating \(\Delta d = 3.00 \times 10^8 \times 0.60 \times 10^{-6}\, = 180\, \text{m}\). This is the additional path length the reflected signal travels compared to the direct signal.

Key Concepts

Signal Delay

When a signal from a TV transmitter takes multiple paths to reach your screen, it can result in ghost images. This happens when one signal travels a longer path and therefore arrives later than the direct signal. The time difference between when the first and second signals arrive is known as "signal delay." Signal delay in broadcasting imagery is crucial because even minute differences can cause visual disturbances. In the given exercise, the delay was 0.60 microseconds. This small delay led to the ghost image appearing to the right of the main image by 1.0 cm due to differing arrival times of signals. Signal delays need to be accounted for in TV technology to ensure clear images: - Ensuring signals travel the shortest path possible - Minimizing reflections by avoiding metallic obstructions Understanding and managing signal delay is vital for maintaining high-quality transmission.

Path Difference

Path difference refers to how far one signal travels compared to another. This comes into play when talking about wave interference, as in the case of ghost TV images. Longer traveled paths cause these signals to arrive fractions of a second later, which can lead to duplicated or shifted images.In our exercise, the reflected signal had a path difference of 180 meters compared to the direct path. This means it traveled 180 meters further before reaching the receiver. To calculate path difference, we use the formula: - \[ \Delta d = c \times \Delta t \] - Here, \( \Delta d \) represents path difference, \( c \) is the speed of light, and \( \Delta t \) is the time delay Knowing the path difference is helpful because engineers can then mitigate or use this difference to improve signal quality.

Speed of Light

Signals, like those in television broadcasting, travel at the speed of light, which is approximately \( 3.00 \times 10^8 \) meters per second. This constant speed is crucial for understanding how signals behave over distance and time. Speed of light plays a key role in assessing signal delay and path differences because it determines how quickly a signal travels from point A to point B. In the exercise, we used the speed of light to calculate how much further the reflected signal traveled compared to the direct signal. By multiplying the speed of light by the signal delay time, we derived that the path difference was 180 meters. This concept is foundational not only in television broadcasting but also in many other technologies, like fiber optics and satellite communication.Understanding the speed of light: - Helps calculate travel time for signals across distances - Allows precise identification of timing differences in overlapping signals Grasping this fundamental physical constant is important for anyone interested in communication technology or physics.