Answer:
62.8 Ī¼C
Explanation:
Here is the complete question
The volume electric charge density of a solid sphere is given by the following equation: Ļ = (0.2 mC/māµ)rĀ²The variable r denotes the distance from the center of the sphere, in spherical coordinates. What is the net electric charge (in Ī¼C) of the sphere if the radius of the sphere is 0.5 m?
Solution
The total charge on the sphere Q = ā«ā«ā«ĻdV where Ļ = volume charge density = 0.2rĀ² and dV = volume element in spherical coordinates = rĀ²sinĪødĪødrdĪ¦
So, Ā Q = Ā ā«ā«ā«ĻdV
Q = Ā ā«ā«ā«ĻrĀ²sinĪødĪødrdĪ¦
Q = Ā ā«ā«ā«(0.2rĀ²)rĀ²sinĪødĪødrdĪ¦
Q = Ā ā«ā«ā«0.2rā“sinĪødĪødrdĪ¦
We integrate from r = 0 to r = 0.5 m, Īø = 0 to Ļ and Ī¦ = 0 to 2Ļ
So, Q = Ā ā«ā«ā«0.2rā“sinĪødĪødrdĪ¦
Q = Ā ā«ā«ā«0.2rā“[ā«sinĪødĪø]drdĪ¦
Q = Ā ā«ā«0.2rā“[-cosĪø]drdĪ¦
Q = Ā ā«ā«0.2rā“-[cosĻ - cos0]drdĪ¦
Q = Ā ā«ā«ā«0.2rā“-[-1 - 1]drdĪ¦
Q = Ā ā«ā«0.2rā“-[- 2]drdĪ¦
Q = Ā ā«ā«0.2rā“(2)drdĪ¦
Q = Ā ā«ā«0.4rā“drdĪ¦
Q = Ā ā«0.4rā“drā«dĪ¦
Q = Ā ā«0.4rā“dr[Ī¦]
Q = Ā ā«0.4rā“dr[2Ļ - 0]
Q = Ā ā«0.4rā“dr[2Ļ]
Q = Ā ā«0.8Ļrā“dr
Q = Ā 0.8Ļā«rā“dr
Q = Ā 0.8Ļ[rāµ/5]
Q = 0.8Ļ[(0.5 m)āµ/5 - (0 m)āµ/5]
Q = 0.8Ļ[0.125 māµ/5 - 0 māµ/5]
Q = 0.8Ļ[0.025 māµ - 0 māµ]
Q = 0.8Ļ[0.025 māµ]
Q = (0.02Ļ mC/māµ) māµ
Q = 0.0628 mC
Q = 0.0628 Ć 10ā»Ā³ C
Q = 62.8 Ć 10ā»Ā³ Ć 10ā»Ā³ C
Q = 62.8 Ć 10ā»ā¶ C
Q = 62.8 Ī¼C
