justynejoseph9914 justynejoseph9914
  • 10-08-2019
  • Mathematics
contestada

Use Gauss's Law to find the charge contained in the solid hemisphere x2 + y2 + z2 ≤ a2, z ≥ 0, if the electric field is E(x, y, z) = x i + y j + 6z k.

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isyllus
isyllus isyllus
  • 13-08-2019

Answer: [tex]\frac{16}{3}[/tex]π[tex]a^{3}[/tex] .

Given:

[tex]x^{2}+y^{2}+z^{2} \leq a^{2}, z \geq 0[/tex]

Using Gauss's Law = ∫∫s E ·dS  

= ∫∫∫ div E dV,  

⇒ Divergence (Gauss') Theorem  

= ∫∫∫ (1+1+6) dV  

= 8×(volume of the hemisphere, radius "a")  

= 8× ([tex]\frac{1}{2}[/tex])(4/3)π[tex]a^{3}[/tex]  

= [tex]\frac{16}{3}[/tex]π[tex]a^{3}[/tex] .

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