The amplitude of upper and lower side bands of A.M. wave where a carrier signal with frequency 11.21 MHz, peak voltage 15 V is amplitude modulated by a 7.7 kHz sine wave of 5V amplitude are `(a)/(10) V and (b)/(10)V` respectively. Then the value of `(a)/(b)` is ____________.

Answer

To solve the problem, we need to find the ratio \( \frac{a}{b} \) of the amplitudes of the upper and lower sidebands of an amplitude modulated (AM) wave.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Carrier frequency \( f_c = 11.21 \, \text{MHz} \)
- Carrier peak voltage \( V_c = 15 \, \text{V} \)
- Modulating frequency \( f_m = 7.7 \, \text{kHz} \)
- Modulating amplitude \( V_m = 5 \, \text{V} \)

2. **Understand the AM Wave**:
- The amplitude of the upper sideband (USB) and lower sideband (LSB) in an AM wave can be expressed as:
\[ A_{USB} = \frac{\mu V_c}{2} \]
\[ A_{LSB} = \frac{\mu V_c}{2} \]
where \( \mu \) is the modulation index.

3. **Calculate the Modulation Index \( \mu \)**:
- The modulation index \( \mu \) is given by:
\[ \mu = \frac{V_m}{V_c} = \frac{5 \, \text{V}}{15 \, \text{V}} = \frac{1}{3} \]

4. **Substitute \( \mu \) into the Sideband Amplitudes**:
- For the upper sideband:
\[ A_{USB} = \frac{\mu V_c}{2} = \frac{\frac{1}{3} \times 15}{2} = \frac{15}{6} = 2.5 \, \text{V} \]
- For the lower sideband:
\[ A_{LSB} = \frac{\mu V_c}{2} = \frac{\frac{1}{3} \times 15}{2} = \frac{15}{6} = 2.5 \, \text{V} \]

5. **Express the Sideband Amplitudes in Terms of \( a \) and \( b \)**:
- From the problem, we have:
\[ A_{USB} = \frac{a}{10} \quad \text{and} \quad A_{LSB} = \frac{b}{10} \]
- Since both amplitudes are equal, we can set:
\[ \frac{a}{10} = 2.5 \quad \text{and} \quad \frac{b}{10} = 2.5 \]

6. **Solve for \( a \) and \( b \)**:
- From \( \frac{a}{10} = 2.5 \):
\[ a = 2.5 \times 10 = 25 \]
- From \( \frac{b}{10} = 2.5 \):
\[ b = 2.5 \times 10 = 25 \]

7. **Calculate the Ratio \( \frac{a}{b} \)**:
- Now, we find:
\[ \frac{a}{b} = \frac{25}{25} = 1 \]

### Final Answer:
The value of \( \frac{a}{b} \) is \( 1 \).