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### Problem Statement: Electric Field Calculation
Consider a thin, spherical shell of radius \( 14.5 \) cm with a total charge of \( 29.9 \) μC distributed uniformly on its surface.
#### (a) Find the electric field 10.0 cm from the center of the charge distribution.
#### (b) Find the electric field 24.5 cm from the center of the charge distribution.
---
### Explanation
To solve these problems, we will apply Gauss's Law, which relates the electric field to the charge enclosed by a Gaussian surface. The electric field \( E \) due to a spherical charge distribution can be calculated differently depending on whether the point of interest is inside or outside the spherical shell:
1. **Inside the Spherical Shell ( \( r < R \) )**:
- For any point inside a uniformly charged spherical shell, the electric field is zero because the charges on the shell produce a net zero electric field inside.
2. **Outside the Spherical Shell ( \( r \geq R \) )**:
- The electric field due to a spherical shell of charge is as if all the charge were concentrated at the center of the shell. Thus, it can be calculated using Coulomb's Law:
\[
E = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2}
\]
where \( \epsilon_0 \) is the permittivity of free space, \( Q \) is the total charge, and \( r \) is the distance from the center of the sphere.
Let's solve the problems step-by-step.
### Solution
#### (a) Electric Field at 10.0 cm from the Center
Since \( 10.0 \) cm is less than the radius \( 14.5 \) cm (i.e., \( r < R \)), according to Gauss's Law, the electric field inside a spherical shell is zero.
\[
E_{\text{inside}} = 0
\]
Thus, the electric field 10.0 cm from the center of the charge distribution is:
\[
\boxed{0}
\]
#### (b) Electric Field at 24.5 cm from the Center
Since \( 24.5 \) cm is greater than the radius \( 14.5 \) cm (i.e., \( r \geq R \)), we](https://content.bartleby.com/qna-images/question/c4571419-cef3-467f-82d2-89ad06819ab1/c6d6f843-6d34-4603-938a-afff07ca06d0/q40gn2g.png)
Transcribed Image Text:---
### Problem Statement: Electric Field Calculation
Consider a thin, spherical shell of radius \( 14.5 \) cm with a total charge of \( 29.9 \) μC distributed uniformly on its surface.
#### (a) Find the electric field 10.0 cm from the center of the charge distribution.
#### (b) Find the electric field 24.5 cm from the center of the charge distribution.
---
### Explanation
To solve these problems, we will apply Gauss's Law, which relates the electric field to the charge enclosed by a Gaussian surface. The electric field \( E \) due to a spherical charge distribution can be calculated differently depending on whether the point of interest is inside or outside the spherical shell:
1. **Inside the Spherical Shell ( \( r < R \) )**:
- For any point inside a uniformly charged spherical shell, the electric field is zero because the charges on the shell produce a net zero electric field inside.
2. **Outside the Spherical Shell ( \( r \geq R \) )**:
- The electric field due to a spherical shell of charge is as if all the charge were concentrated at the center of the shell. Thus, it can be calculated using Coulomb's Law:
\[
E = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2}
\]
where \( \epsilon_0 \) is the permittivity of free space, \( Q \) is the total charge, and \( r \) is the distance from the center of the sphere.
Let's solve the problems step-by-step.
### Solution
#### (a) Electric Field at 10.0 cm from the Center
Since \( 10.0 \) cm is less than the radius \( 14.5 \) cm (i.e., \( r < R \)), according to Gauss's Law, the electric field inside a spherical shell is zero.
\[
E_{\text{inside}} = 0
\]
Thus, the electric field 10.0 cm from the center of the charge distribution is:
\[
\boxed{0}
\]
#### (b) Electric Field at 24.5 cm from the Center
Since \( 24.5 \) cm is greater than the radius \( 14.5 \) cm (i.e., \( r \geq R \)), we
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