A breathing sphere of radius 0.1m radiates spherical waves into air at 100Hz with an intensity of 200mW/m2 at the distance 0.5m from the center of the sphere. In this case,
1) Compute the radiated acoustic power.
2) Compute the intensity at the surface of the sphere.
3) Compute the intensity at the distance 1m from the center.
(1) 200π mW (2) 10W/m2 (3) 50mW/m2
(1) 400π mW (2) 5W/m2 (3) 100mW/m2
(1) 100π mW (2) 10W/m2 (3) 25mW/m2
(1) 200π mW (2) 5W/m2 (3) 200mW/m2
(1) 200π mW (2) 5W/m2 (3) 50mW/m2
To solve this problem, we can use the inverse square law which states that the intensity decreases with the square of the distance from the source.
1) The radiated acoustic power is the total amount of power radiated by the sphere. The formula to calculate the radiated power is given by:
Radiated power = Intensity × Surface area
The surface area of a sphere is given by:
Surface area = 4πr^2
where r is the radius of the sphere. Substituting the given values:
Surface area = 4π(0.1^2)
= 0.04π m^2
Now, substituting the values into the radiated power formula:
Radiated power = 200 mW/m^2 × 0.04π m^2
= 8π mW
≈ 25.13 mW
2) The intensity at the surface of the sphere is the intensity of the waves at the same distance as the radius of the sphere. So, at r = 0.1m, the intensity is given as 200 mW/m^2.
3) Using the inverse square law, we can find the intensity at a distance of 1m from the center of the sphere. The formula to calculate the intensity at a distance is:
Intensity at distance 1 = Intensity at distance 0.5 × (Distance at distance 0.5 / Distance at distance 1)^2
Substituting the given values:
Intensity at distance 1 = 200 mW/m^2 × (0.5 / 1)^2
= 200 mW/m^2 × 0.25
= 50 mW/m^2
Therefore, the correct answers are:
(1) 200π mW
(2) 5W/m^2
(3) 50mW/m^2
To find the answers to the given questions, we need to use the formulas related to acoustic power and intensity in spherical waves. Let's go step by step:
1) Compute the radiated acoustic power:
The radiated acoustic power, in watts, can be obtained using the formula:
Power = 4πr^2 * I
where r is the distance from the source and I is the intensity of the sound wave at that distance.
In this case, the intensity is given as 200 mW/m^2 at a distance of 0.5m from the center of the sphere. So, substituting the values into the formula:
Power = 4π(0.5)^2 * 0.2 = 0.4π W ≈ 1.2566 W
Therefore, the radiated acoustic power is approximately 1.2566 W.
2) Compute the intensity at the surface of the sphere:
The intensity at the surface of the sphere, denoted by Is, can be calculated using the formula:
Is = (Power) / (4πr^2)
where r is the radius of the sphere.
In this case, the radius of the sphere is given as 0.1m, so substituting the values:
Is = 1.2566 / (4π(0.1)^2) = 10 W/m^2
Therefore, the intensity at the surface of the sphere is 10 W/m^2.
3) Compute the intensity at the distance 1m from the center:
To find the intensity at a distance of 1m from the center, we can again use the formula:
I = (Power) / (4πr^2)
where r is the distance from the center of the sphere.
In this case, the distance from the center is given as 1m, so substituting the values:
I = 1.2566 / (4π(1)^2) = 0.1 W/m^2 = 100 mW/m^2
Therefore, the intensity at a distance of 1m from the center is 100 mW/m^2.
So, the correct option among the given choices is:
(1) 200π mW (2) 10W/m^2 (3) 100mW/m^2