Expensive amplifier A is rated at 250 W, while the more modest amplifier B is rated at 45 W. (a) Estimate the sound level in decibels you would expect at a point 3.5 m from a loudspeaker connected in turn to each amp. (b) Will the expensive amp sound twice as loud as the cheaper one?
To estimate the sound level in decibels for each amplifier at a point 3.5 m away from the loudspeaker, we need to consider the speaker's power and the distance.
The sound level in decibels (dB) can be calculated using the formula:
L = 10 * log10(P / P₀)
where L is the sound level in decibels, P is the power of the sound source, and P₀ is the reference power (usually taken as the threshold of human hearing, which is approximately 1 picowatt or 10^(-12) watts).
(a) For amplifier A with a power rating of 250 W:
L₁ = 10 * log10(250 / P₀)
Similarly, for amplifier B with a power rating of 45 W:
L₂ = 10 * log10(45 / P₀)
To find the difference in sound levels, we can subtract L₂ from L₁:
ΔL = L₁ - L₂
(b) To determine whether the expensive amplifier will sound twice as loud as the cheaper one, we need to compare the sound level difference to the threshold of human perception.
Let's go through the calculation step-by-step:
Step 1: Finding the reference power (P₀):
P₀ = 10^(-12) watts
Step 2: Calculating the sound level for amplifier A (L₁):
L₁ = 10 * log10(250 / P₀)
Step 3: Calculating the sound level for amplifier B (L₂):
L₂ = 10 * log10(45 / P₀)
Step 4: Calculating the difference in sound levels (ΔL):
ΔL = L₁ - L₂
Step 5: Determining the perception threshold:
The threshold of human perception is approximately 1 dB.
If the ΔL is equal to or greater than 1 dB, then the expensive amplifier will sound noticeably louder.
If the ΔL is less than 1 dB, the difference in loudness may not be noticeable to most people.
By following these steps and plugging in the provided power ratings, you can estimate the sound levels in decibels and compare the loudness of the two amplifiers.