The intensity of a wave emitted from a small source at a distance 10 m is `20 xx 10^(-3)"Wm"^(-2).` What is the intensity at a distance 100 m ?

Answer

To solve the problem of finding the intensity of a wave at a distance of 100 m, given that the intensity at a distance of 10 m is \(20 \times 10^{-3} \, \text{W/m}^2\), we can follow these steps:

### Step 1: Understand the relationship between intensity and distance
Intensity \(I\) of a wave emitted from a point source is inversely proportional to the square of the distance \(R\) from the source. This can be expressed mathematically as:
\[ I \propto \frac{1}{R^2} \]

### Step 2: Set up the equation for the two distances
Let \(I_1\) be the intensity at distance \(R_1\) (10 m) and \(I_2\) be the intensity at distance \(R_2\) (100 m). According to the relationship:
\[ \frac{I_1}{I_2} = \frac{R_2^2}{R_1^2} \]

### Step 3: Substitute the known values
We know:
- \(I_1 = 20 \times 10^{-3} \, \text{W/m}^2\)
- \(R_1 = 10 \, \text{m}\)
- \(R_2 = 100 \, \text{m}\)

Substituting these values into the equation:
\[ \frac{20 \times 10^{-3}}{I_2} = \frac{(100)^2}{(10)^2} \]

### Step 4: Calculate the ratio of distances
Calculating the ratio:
\[ \frac{(100)^2}{(10)^2} = \frac{10000}{100} = 100 \]

### Step 5: Solve for \(I_2\)
Now we can rearrange the equation to solve for \(I_2\):
\[ I_2 = \frac{20 \times 10^{-3}}{100} \]

### Step 6: Perform the calculation
Calculating \(I_2\):
\[ I_2 = 20 \times 10^{-3} \div 100 = 0.2 \times 10^{-3} \, \text{W/m}^2 \]

### Step 7: Final answer
Thus, the intensity at a distance of 100 m is:
\[ I_2 = 0.2 \times 10^{-3} \, \text{W/m}^2 \]