Ratio of amplitude for two wave is 1:4 .Find the ratio of maximum intensity to minimum intensity.

Answer

To solve the problem of finding the ratio of maximum intensity to minimum intensity given the ratio of amplitudes of two waves is 1:4, we can follow these steps:

### Step 1: Understand the relationship between amplitude and intensity
The intensity (I) of a wave is proportional to the square of its amplitude (A). Therefore, if we have two waves with amplitudes A1 and A2, the relationship can be expressed as:
\[ I \propto A^2 \]

### Step 2: Write down the given data
From the problem, we know:
- The ratio of amplitudes is given as:
\[ \frac{A_1}{A_2} = \frac{1}{4} \]

### Step 3: Calculate the intensities
Using the relationship between intensity and amplitude, we can express the intensities as:
\[ I_1 \propto A_1^2 \]
\[ I_2 \propto A_2^2 \]

Substituting the ratio of amplitudes:
- Let \( A_1 = 1 \) and \( A_2 = 4 \).
- Then:
\[ I_1 = A_1^2 = 1^2 = 1 \]
\[ I_2 = A_2^2 = 4^2 = 16 \]

### Step 4: Calculate maximum intensity (Imax)
The maximum intensity (Imax) when two waves interfere constructively is given by:
\[ I_{max} = ( \sqrt{I_1} + \sqrt{I_2} )^2 \]
Substituting the values of I1 and I2:
\[ I_{max} = ( \sqrt{1} + \sqrt{16} )^2 \]
\[ I_{max} = ( 1 + 4 )^2 = 5^2 = 25 \]

### Step 5: Calculate minimum intensity (Imin)
The minimum intensity (Imin) when two waves interfere destructively is given by:
\[ I_{min} = ( \sqrt{I_1} - \sqrt{I_2} )^2 \]
Substituting the values of I1 and I2:
\[ I_{min} = ( \sqrt{1} - \sqrt{16} )^2 \]
\[ I_{min} = ( 1 - 4 )^2 = (-3)^2 = 9 \]

### Step 6: Calculate the ratio of maximum intensity to minimum intensity
Now we can find the ratio of maximum intensity to minimum intensity:
\[ \text{Ratio} = \frac{I_{max}}{I_{min}} = \frac{25}{9} \]

### Final Answer
The ratio of maximum intensity to minimum intensity is:
\[ \frac{25}{9} \]

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