Television signals reach us only through ground waves. The range R is related to the height h of the transmitter antenna as

Options

A. `R alpha h^(1//3)`
B. `R alpha h^(1//2)`
C. `R alpha h^2`
D. `R alpha h`

Explanation

To solve the problem of how the range \( R \) of television signals relates to the height \( h \) of the transmitter antenna, we can follow these steps:

### Step 1: Understand the Geometry
We start by visualizing the scenario. We have the Earth, and at a height \( h \) above the Earth's surface, there is a transmitter antenna. The radius of the Earth can be denoted as \( r \).

### Step 2: Set Up the Right Triangle
The situation can be represented as a right triangle:
- One leg of the triangle is the radius of the Earth, \( r \).
- The other leg is the height of the antenna, \( h \).
- The hypotenuse of the triangle is \( h + r \) (the distance from the antenna to the horizon).

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem, we can write:
\[ R^2 + r^2 = (h + r)^2 \]
where \( R \) is the range of the signal.

### Step 4: Expand the Equation
Expanding the right side gives:
\[ R^2 + r^2 = h^2 + 2hr + r^2 \]
We can simplify this by subtracting \( r^2 \) from both sides:
\[ R^2 = h^2 + 2hr \]

### Step 5: Neglect the \( h^2 \) Term
Since the height \( h \) of the antenna is very small compared to the radius of the Earth \( r \) (i.e., \( h \ll r \)), we can neglect the \( h^2 \) term:
\[ R^2 \approx 2hr \]

### Step 6: Solve for \( R \)
Taking the square root of both sides gives:
\[ R \approx \sqrt{2hr} \]

### Step 7: Establish the Relationship
From this equation, we can see that the range \( R \) is directly proportional to the square root of the height \( h \):
\[ R \propto \sqrt{h} \]

### Conclusion
Thus, the relationship between the range \( R \) and the height \( h \) of the transmitter antenna is:
\[ R \propto h^{1/2} \]