College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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![### Problem Statement
A uniformly charged disk with radius \( R = 40.0 \) cm and uniform charge density \( \sigma = 8.90 \times 10^{-3} \) C/m² lies in the xy-plane, with its center at the origin. What is the electric field (in MN/C) due to the charged disk at the following locations?
#### (a) \( z = 5.00 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
#### (b) \( z = 10.0 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
#### (c) \( z = 50.0 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
#### (d) \( z = 200 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
### Explanation for Educational Website
This problem involves calculating the electric field generated by a uniformly charged disk at different points along the z-axis.
#### Definitions:
- **Radius (R):** The distance from the center of the disk to its edge.
- **Uniform charge density (\(\sigma\)):** Charge per unit area on the surface of the disk.
- **xy-plane:** A two-dimensional plane determined by the x-axis and y-axis.
#### Strategy:
Use the formula for the electric field due to a uniformly charged disk at a point along its perpendicular axis (the z-axis):
\[ E(z) = \frac{\sigma}{2 \epsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right) \]
where:
- \( z \) is the distance from the disk along the z-axis.
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.85 \times 10^{-12} \text{C}^2/(\text{N} \cdot \text{m}^2) \)).
#### Steps:
1. **Convert cm to m:** Ensure all distances are in meters for consistency in units.
2. **Substitute values into the formula:** Plug in the radius \( R \), charge density \( \sigma \), and distance \( z \)](https://content.bartleby.com/qna-images/question/0d33c096-b905-4215-971d-b02ba4b92127/84004de1-8198-4823-8d56-e75763b30291/9eeh2dn_processed.png)
Transcribed Image Text:### Problem Statement
A uniformly charged disk with radius \( R = 40.0 \) cm and uniform charge density \( \sigma = 8.90 \times 10^{-3} \) C/m² lies in the xy-plane, with its center at the origin. What is the electric field (in MN/C) due to the charged disk at the following locations?
#### (a) \( z = 5.00 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
#### (b) \( z = 10.0 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
#### (c) \( z = 50.0 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
#### (d) \( z = 200 \) cm
\[ \boxed{\hspace{3cm}} \text{MN/C} \]
### Explanation for Educational Website
This problem involves calculating the electric field generated by a uniformly charged disk at different points along the z-axis.
#### Definitions:
- **Radius (R):** The distance from the center of the disk to its edge.
- **Uniform charge density (\(\sigma\)):** Charge per unit area on the surface of the disk.
- **xy-plane:** A two-dimensional plane determined by the x-axis and y-axis.
#### Strategy:
Use the formula for the electric field due to a uniformly charged disk at a point along its perpendicular axis (the z-axis):
\[ E(z) = \frac{\sigma}{2 \epsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right) \]
where:
- \( z \) is the distance from the disk along the z-axis.
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.85 \times 10^{-12} \text{C}^2/(\text{N} \cdot \text{m}^2) \)).
#### Steps:
1. **Convert cm to m:** Ensure all distances are in meters for consistency in units.
2. **Substitute values into the formula:** Plug in the radius \( R \), charge density \( \sigma \), and distance \( z \)
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