27: The Electric Field of a Ball of Uniform Charge Density

INTRO:
A solid ball of radius r_b has a uniform charge density ρ.
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PART A:
What is the magnitude of the electric field E(r) at a distance r>r_b from the center of the ball?
Express your answer in terms of ρ, r_b, r, and ϵ₀.

SOLUTION:
Since a charge density is given, the charge must be acquired from this density.
A density has the units [something] per volume. Therefore, the charge can be determined if the volume of the sphere is determined first. 
V = (4/3)πr³
∴ q = ρ⋅V = ρ⋅(4/3)πr³

E(r) = ρr_b³ / (3ε₀⋅r²)

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PART B:
What is the magnitude of the electric field E(r) at a distance r<r_b from the center of the ball?
Express your answer in terms of ρ, r_b, r, and ϵ₀.

SOLUTION:
Since the Gaussian surface is inside the ball, the surface encloses only a fraction of the ball's charge. This is why Gauss's law is so useful: As long as it is symmetrically distributed, the charge outside the Gaussian surface is irrelevant in calculating the field!

The net charge enclosed by the Gaussian surface is different than that of the net charge enclosed in part A.

E(r) = ρr / (3ε₀)

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PART C:

Let E(r) represent the electric field due to the charged ball throughout all of space. Which of the following statements about the electric field are true?
Check all that apply.

• E(0) = 0
• E(r_b)=0
• lim_r→∞E(r)=0
• The maximum electric field occurs when r=0
• The maximum electric field occurs when r=r_b
• The maximum electric field occurs as r→∞.

SOLUTION:

E(0) = 0
lim_r→∞E(r)=0
The maximum electric field occurs when r=r_b

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