Three waves producing displacement in the same direction of same frequency and of amplitudes `10etam, 4etam" and "7eta` m arrive at a point with successive phase difference of `pi//2`. The amplitude of the resultant wave is :--

Options

A. `2etam`
B. `7etam`
C. `5etam`
D. 1

Answer

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

### Step 1: Identify the given parameters
We have three waves with the following amplitudes:
- \( A_1 = 10 \, \text{nm} \)
- \( A_2 = 4 \, \text{nm} \)
- \( A_3 = 7 \, \text{nm} \)

The phase differences between the waves are:
- The phase difference between the first and second wave is \( \frac{\pi}{2} \).
- The phase difference between the second and third wave is also \( \frac{\pi}{2} \).

### Step 2: Represent the waves as phasors
We can represent the waves as phasors in a complex plane:
- The first wave (amplitude \( A_1 \)) can be represented as \( 10 \, \text{nm} \) at an angle of \( 0 \) radians.
- The second wave (amplitude \( A_2 \)) can be represented as \( 4 \, \text{nm} \) at an angle of \( \frac{\pi}{2} \) radians.
- The third wave (amplitude \( A_3 \)) can be represented as \( 7 \, \text{nm} \) at an angle of \( \pi \) radians.

### Step 3: Calculate the components of each wave
We can break down each wave into its x (real) and y (imaginary) components:
- For \( A_1 \):
- \( x_1 = 10 \cos(0) = 10 \)
- \( y_1 = 10 \sin(0) = 0 \)

- For \( A_2 \):
- \( x_2 = 4 \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( y_2 = 4 \sin\left(\frac{\pi}{2}\right) = 4 \)

- For \( A_3 \):
- \( x_3 = 7 \cos(\pi) = -7 \)
- \( y_3 = 7 \sin(\pi) = 0 \)

### Step 4: Sum the components
Now, we sum the x and y components:
- Total x-component:
\[ X = x_1 + x_2 + x_3 = 10 + 0 - 7 = 3 \]

- Total y-component:
\[ Y = y_1 + y_2 + y_3 = 0 + 4 + 0 = 4 \]

### Step 5: Calculate the resultant amplitude
The resultant amplitude \( A \) can be found using the Pythagorean theorem:
\[ A = \sqrt{X^2 + Y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{nm} \]

### Conclusion
The amplitude of the resultant wave is \( 5 \, \text{nm} \).

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