Question

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.

Answer 1

Answer: $\frac{16}{3}$π$a^{3}$ .

Given:

$x^{2}+y^{2}+z^{2} \leq a^{2}, z \geq 0$

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× ($\frac{1}{2}$)(4/3)π$a^{3}$  

= $\frac{16}{3}$π$a^{3}$ .