An electromagnetic wave emitted by source travels 21 km to arrive at a receiver. The wave while travelling in another path is reflected from a surface at 19 km away and further travels 12 km to reach the same receiver. If destructive interference occurs at the receiving end, the mximum wavelength of the wave is

Options

A. `0.5 km`
B. `1 km`
C. `5 km`
D. `10 km`

Answer

To solve the problem, we need to determine the maximum wavelength of the electromagnetic wave that results in destructive interference at the receiver. Here are the steps to arrive at the solution:

### Step 1: Identify the paths of the electromagnetic wave
- The first path (X1) is the direct path from the source to the receiver, which is given as 21 km.
- The second path (X2) involves the wave being reflected from a surface. The distance to the reflecting surface is 19 km, and then it travels an additional 12 km to reach the receiver.

### Step 2: Calculate the total distance for the second path
- The total distance for the second path (X2) can be calculated as:
\[ X2 = 19 \text{ km} + 12 \text{ km} = 31 \text{ km} \]

### Step 3: Determine the path difference
- The path difference (ΔX) between the two paths is given by:
\[ \Delta X = X2 - X1 = 31 \text{ km} - 21 \text{ km} = 10 \text{ km} \]

### Step 4: Apply the condition for destructive interference
- The condition for destructive interference is modified when one of the waves is reflected. The modified condition is:
\[ \lambda m = \Delta X \]
where \( m \) is an integer (0, 1, 2, ...).

### Step 5: Find the maximum wavelength
- To find the maximum wavelength (λ_max), we can set \( m = 1 \) (the first order of destructive interference):
\[ \lambda_{\text{max}} = \Delta X = 10 \text{ km} \]

### Conclusion
- The maximum wavelength of the wave that results in destructive interference at the receiving end is **10 kilometers**.