Question

Question: (III) (a) Expensive amplifier A is rated at 220 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?

Short answer

(a) The sound level from amplifier A and B are \(121.55\;{\rm{dB}}\) and \(114.77\;{\rm{dB}}\) respectively. (b) The expensive amp will not sound twice as loud as the cheaper one.

Step-by-step solution

Understanding the relation of sound intensity with sound power

The intensity of the sound is directly proportional to the sound power and it is inversely proportional to the distance of source and receiver.

Given data

The power of amplifier A is \({P_A} = 220\;{\rm{W}}\).

The power of amplifier B is \({P_B} = 45\;{\rm{W}}\).

The distance of point from loudspeaker is \(r = 3.5\;{\rm{m}}\).

The standard value for the threshold intensity is \({I_0} = 1.0 \times {10^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}\).

Evaluating the sound intensity and sound level produced by amplifier A

The intensity of sound by amplifier A is calculated below:

\(\begin{array}{c}{I_A} = \frac{{{P_A}}}{A}\\ = \frac{{{P_A}}}{{\left( {4\pi {r^2}} \right)}}\end{array}\)

Here, A is the area.

Substitute the values in the above equation.

\(\begin{array}{c} = \frac{{220\;{\rm{W}}}}{{4\pi {{\left( {3.5\;{\rm{m}}} \right)}^2}}}\\ = 1.43\;{\rm{W/}}{{\rm{m}}^2}\end{array}\)

The sound level from amplifier A is calculated below:

\({\beta _A} = 10\log \left( {\frac{{{I_A}}}{{{I_0}}}} \right)\)

Substitute the values in the above equation.

\(\begin{array}{c}{\beta _A} = 10\log \left( {\frac{{1.43\;{\rm{W/}}{{\rm{m}}^2}}}{{1.0 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right)\\{\beta _A} = 121.55\;{\rm{dB}}\end{array}\)

Evaluating the sound intensity and sound level produced by amplifier B

The intensity of sound by amplifier B is calculated below:

\(\begin{array}{c}{I_B} = \frac{{{P_B}}}{A}\\ = \frac{{{P_B}}}{{\left( {4\pi {r^2}} \right)}}\end{array}\)

Substitute the values in the above equation.

\(\begin{array}{c}{I_B} = \frac{{45\;{\rm{W}}}}{{4\pi {{\left( {3.5\;{\rm{m}}} \right)}^2}}}\\{I_B} = 0.30\;{\rm{W/}}{{\rm{m}}^2}\end{array}\)

The sound level from amplifier B is calculated below:

\({\beta _B} = 10\log \left( {\frac{{{I_B}}}{{{I_0}}}} \right)\)

Substitute the values in the above equation.

\(\begin{array}{c}{\beta _B} = 10\log \left( {\frac{{0.30\;{\rm{W/}}{{\rm{m}}^2}}}{{1.0 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right)\\{\beta _B} = 114.77\;{\rm{dB}}\end{array}\)

Hence, the sound level from amplifier A and B are \(121.55\;{\rm{dB}}\) and \(114.77\;{\rm{dB}}\) respectively.

Evaluating that expensive amplifier does not increase loudness of sound

The loudness of sound is independent of the cost of the amplifier. Therefore, the expensive amplifier does not increase the loudness of sound. Hence, the expensive amplifier will not sound twice as loud as the cheaper one.