A message signal of frequency 3 kHz is used to modulate a carrier signal of frequency 1.5 MHz. The bandwidth of the amplitude modulated wave is

We need to find the bandwidth of an amplitude modulated (AM) wave when a message signal of frequency 3 kHz modulates a carrier signal of frequency 1.5 MHz.

In amplitude modulation, the message signal (also called the modulating signal) modulates the amplitude of a high-frequency carrier wave. When a carrier of frequency $$f_c$$ is modulated by a message signal of frequency $$f_m$$, the resulting AM wave contains three frequency components.
The carrier frequency: $$f_c$$
The upper sideband: $$f_c + f_m$$
The lower sideband: $$f_c - f_m$$

The bandwidth of the AM wave is the difference between the highest and lowest frequencies present in the signal, given by $$\text{Bandwidth} = (f_c + f_m) - (f_c - f_m) = 2f_m$$. Notice that the bandwidth depends only on the message signal frequency and is independent of the carrier frequency.

First, we identify the given values: message signal frequency $$f_m = 3$$ kHz and carrier signal frequency $$f_c = 1.5$$ MHz.

Next, substituting into the formula yields $$\text{Bandwidth} = 2f_m = 2 \times 3 \text{ kHz} = 6 \text{ kHz}$$.

To verify, we list the frequency components: upper sideband $$f_c + f_m = 1.5 \text{ MHz} + 3 \text{ kHz} = 1.503 \text{ MHz}$$ and lower sideband $$f_c - f_m = 1.5 \text{ MHz} - 3 \text{ kHz} = 1.497 \text{ MHz}$$. Calculating the bandwidth gives $$1.503 - 1.497 = 0.006 \text{ MHz} = 6 \text{ kHz}$$, which confirms our calculation.

Therefore, the correct answer is Option 2: 6 kHz.