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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27: Problem 27.47

INTRO:
The three parallel planes of charge shown in the figure (Figure 1) have surface charge densities −1/2 η, η, and −1/2 η.

Figure 1

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PART A:
Find the magnitude of the electric field in region 1.

SOLUTION:
<< explanation to be added >>
0 η/ε₀

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PART B:

What is the direction of the electric field in region 1?

• Upward
• Downward
• The field is zero

SOLUTION:
<< explanation to be added >>
The field is zero

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PART C:

Find the magnitude of the electric field in region 2.

SOLUTION:

<< explanation to be added >>

0.500 η/ε₀

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PART D:

What is the direction of the electric field in region 2?

• Upward
• Downward
• The field is zero

SOLUTION:

<< explanation to be added >>

Upward

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PART E:

Find the magnitude of the electric field in region 3.

SOLUTION:

<< explanation to be added >>

0.500 η/ε₀

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PART F:

What is the direction of the electric field in region 3?

• Upward
• Downward
• The field is zero

SOLUTION:

<< explanation to be added >>

Downward

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PART G:

Find the magnitude of the electric field in region 4.

SOLUTION:

<< explanation to be added >>

0 η/ε₀

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PART H:

What is the direction of the electric field in region 4?

• Upward
• Downward
• The field is zero

SOLUTION:

<< explanation to be added >>

The field is zero

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27: Problem 27.46

INTRO:
A uniformly charged ball of radius a and charge −Q is at the center of a hollow metal shell with inner radius b and outer radius c. The hollow sphere has net charge +2Q.


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PART A: 
Determine the magnitude of the electric field in the region r≤a. Give your answer as a multiple of Q/ε₀.
Express your answer in terms of some or all of the variables a, b, c, r, and the constant π.

SOLUTION:
A_sphere = 4πr²

Φ_E = ∫E⋅dA = q/ε₀
E = q/(A⋅ε₀) = q/(4πr²ε₀)

for r≤a, q_in = -Q
E = -r/(4πa³)⋅Q/ε₀ r^

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PART B:
Determine the magnitude of the electric field in the region a<r<b. Give your answer as a multiple of Q/ε₀.
Express your answer in terms of some or all of the variables a, b, c, r, and the constant π.

SOLUTION:
<< explanation to be added >>
E = -1/(4πr²)⋅Q/ε₀ r^

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PART C:

Determine the magnitude of the electric field in the region b<r<c. Give your answer as a multiple of Q/ε₀.

Express your answer in terms of some or all of the variables a, b, c, r, and the constant π.

SOLUTION:

<< explanation to be added >>

0

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PART D:

Determine the magnitude of the electric field in the region c<r. Give your answer as a multiple of Q/ε₀.

Express your answer in terms of some or all of the variables a, b, c, r, and the constant π.

SOLUTION:

<< explanation to be added >>

E = 1/(4πr²)⋅Q/ε₀ r^

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