Question

The maximum distance a up to which T.V transmission from a T.V. tower of height \(h\) can be received is Proportional to (A) \(\mathrm{h}^{(1 / 2)}\) (B) \(\mathrm{h}\) (C) \(\mathrm{h}^{(3 / 2)}\) (D) \(\mathrm{h}^{2}\)

Short answer

The maximum distance up to which TV transmission from a TV tower of height \(h\) can be received is proportional to \(h^{(1/2)}\).

Step-by-step solution

Understand the problem situation

A transmitter sends signals from the top of a tower of height h. Signals can be received in a direct line of sight before being obstructed by the Earth's curve. Let d be the maximum distance of reception, we'll find the proportionality between d and h.

Set up the formula for line of sight distance

The formula for the distance between the line of sight of two objects with given heights is: \(d = \sqrt{2Rh_1} +\sqrt{2Rh_2}\) In our case, the receiver is on the ground (height \(h_2\) = 0) and the transmitter is on the tower (height \(h_1\) = h), thus the formula becomes \(d = \sqrt{2Rh}\), where R is the radius of Earth.

Find the proportionality between d and h

Now we have the formula \(d = \sqrt{2Rh}\). To find the proportionality between d and h, we can rearrange the formula: \(d = \sqrt{2R} * \sqrt{h}\) We can see that d is proportional to \(\sqrt{h}\)

Choose the correct answer

The correct answer is the one that represents the proportionality between d and h: \(d \propto h^{(1/2)}\) Therefore, the correct answer is option (A): \(h^{(1/2)}\)

Key concepts

TV Transmission

Television (TV) transmission is a fascinating concept that involves sending TV signals from a broadcast tower to your home. These signals usually contain both audio and visual information, allowing us to enjoy various TV programs.
TV towers are crucial because they enable wide distribution of signals over long distances. The height of the TV tower is particularly important because it determines how far the signal can travel before the curvature of the Earth blocks it. The higher the tower, the longer the signal can travel without interruption.
This is why you'll often see incredibly tall towers used for TV broadcast, especially in flat areas where there aren't many natural structures to help relay the signal. In terms of its role in ensuring wide coverage, a taller tower would mean that even distant viewers can receive clear TV signals as the line of sight distance – the maximum distance the signal can travel without being obstructed – increases with tower height.

Signal Propagation

Signal propagation refers to the way radio waves travel from the TV tower to the receivers in your homes. Think of it as the journey a signal takes as it moves through space.
These signals travel in straight lines, which means they can be blocked by any large object in their path, like hills or buildings. In the context of TV transmission, the signals are typically in the form of electromagnetic waves, which propagate through the atmosphere effectively under ideal conditions.
An important aspect of signal propagation is the phenomenon of line of sight. This means that for a broadcast signal to reach a receiver, there must be a clear, unobstructed path between the two. This is heavily influenced by both the height of the transmission tower and the local topography.

• High-frequency signals: Usually follow line of sight and can pass through smaller obstacles.
• Low-frequency signals: Can diffract around obstacles to some extent but rely more on line of sight for clear transmission.

Understanding how signals propagate helps in optimizing the placement and height of TV towers to ensure maximum coverage.

Earth's Curvature

The Earth's curvature is a critical factor in the design and function of TV broadcast systems. Because the Earth is round, the surface gradually curves away from a straight line path.
This curving surface means that a signal traveling in a straight line from a TV tower will eventually be blocked by this curvature if the tower is not tall enough. This is why you'll often see exceptionally tall TV towers, which allow the signals to travel farther before they are eclipsed by the Earth's curvature.
The formula for the line of sight distance takes this curvature into account. If you want to find the maximum distance a signal can travel, you'd use the formula: \[ d = \sqrt{2Rh} \] where \( R \) is the Earth's radius and \( h \) is the height of the tower. This formula shows that as the height of the tower increases, the line of sight distance increases, allowing the signal to cover more ground. Understanding the Earth's curvature is thus vital for efficiently planning the placement of TV transmission towers.