Three waves of equal frequency having amplitudes `10mum`, `4mum`, `7mum` arrive at a given point with successive phase difference of `pi//2`, the amplitude of the resulting wave in `mum` is given by

Options

A. 10
B. 5
C. 15
D. 20

Explanation

To find the amplitude of the resulting wave from three waves with given amplitudes and phase differences, we can follow these steps:

### Step 1: Identify the amplitudes and phase differences
We have three waves with the following amplitudes:
- \( A_1 = 10 \, \mu m \)
- \( A_2 = 4 \, \mu m \)
- \( A_3 = 7 \, \mu m \)

The phase differences between the waves are:
- Between \( A_1 \) and \( A_2 \): \( \frac{\pi}{2} \)
- Between \( A_2 \) and \( A_3 \): \( \frac{\pi}{2} \)

### Step 2: Represent the waves as phasors
We can represent each wave as a vector (phasor) in the complex plane:
- The first wave \( A_1 \) can be represented as \( 10 \, \mu m \) along the x-axis: \( (10, 0) \).
- The second wave \( A_2 \) will be at a phase of \( \frac{\pi}{2} \), which means it will be along the y-axis: \( (0, 4) \).
- The third wave \( A_3 \) will be at a phase of \( \pi \) (which is \( \frac{\pi}{2} + \frac{\pi}{2} \)), meaning it will be in the negative x-direction: \( (-7, 0) \).

### Step 3: Calculate the resultant vector
Now, we can find the resultant of these three vectors.

1. **Combine the x-components:**
- From \( A_1 \): \( 10 \)
- From \( A_3 \): \( -7 \)
- Total x-component: \( 10 - 7 = 3 \, \mu m \)

2. **Combine the y-components:**
- From \( A_2 \): \( 4 \)
- Total y-component: \( 4 \, \mu m \)

### Step 4: Use the Pythagorean theorem to find the resultant amplitude
The resultant amplitude \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2} \]
Substituting the values:
\[ R = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \mu m \]

### Final Answer
The amplitude of the resulting wave is \( 5 \, \mu m \).

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