If a carrier wave $$c(t) = A\sin\omega_c t$$ is amplitude modulated by a modulator signal $$m(t) = A\sin\omega_m t$$ then the equation of modulated signal $$[C_m(t)]$$ and its modulation index are respectively

We are given a carrier wave $$ c(t) = A \sin \omega_c t $$ and a modulating signal $$ m(t) = A \sin \omega_m t $$. We need to find the equation of the modulated signal $$ C_m(t) $$ and its modulation index.

In amplitude modulation (AM), the modulated signal is formed by varying the amplitude of the carrier wave in proportion to the modulating signal. The general form of an AM wave is:

$$ C_m(t) = [A_c + m(t)] \sin \omega_c t $$

Here, $$ A_c $$ is the amplitude of the carrier wave. From the carrier wave equation $$ c(t) = A \sin \omega_c t $$, we see that $$ A_c = A $$. The modulating signal is $$ m(t) = A \sin \omega_m t $$. Substituting these into the AM equation:

$$ C_m(t) = [A + A \sin \omega_m t] \sin \omega_c t $$

Factor out $$ A $$ from the terms inside the brackets:

$$ C_m(t) = A \left(1 + \sin \omega_m t\right) \sin \omega_c t $$

So, the equation of the modulated signal is $$ C_m(t) = A (1 + \sin \omega_m t) \sin \omega_c t $$.

Next, we find the modulation index, denoted by $$ \mu $$. The modulation index is defined as the ratio of the amplitude of the modulating signal to the amplitude of the carrier wave:

$$ \mu = \frac{\text{Amplitude of modulating signal}}{\text{Amplitude of carrier wave}} $$

The modulating signal is $$ m(t) = A \sin \omega_m t $$, so its amplitude is $$ A $$. The carrier wave is $$ c(t) = A \sin \omega_c t $$, so its amplitude is also $$ A $$. Therefore:

$$ \mu = \frac{A}{A} = 1 $$

So, the modulation index is 1.

Now, comparing with the options:

A. $$ C_m(t) = A(1 + \sin\omega_m t)\sin\omega_c t $$ and 2 → Incorrect modulation index

B. $$ C_m(t) = A(1 + \sin\omega_m t)\sin\omega_m t $$ and 1 → Incorrect carrier frequency term

C. $$ C_m(t) = A(1 + \sin\omega_m t)\sin\omega_c t $$ and 1 → Matches our result

D. $$ C_m(t) = A(1 + \sin\omega_c t)\sin\omega_m t $$ and 2 → Incorrect frequencies and modulation index

Hence, the correct answer is Option C.