College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![## Calculating Electric Potential from Electric Field
To find the electric potential from the electric field, we can start by understanding the given equation. Here is the specific equation for the electric field \( E \):
\[ E = \frac{\sigma}{2\epsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right) \]
Where:
- \( \sigma \) is the surface charge density.
- \( \epsilon_0 \) is the permittivity of free space.
- \( z \) is the distance from the charge.
- \( R \) is a given constant (possibly radius).
### Explanation of the Formula
1. **Electric Field (\( E \))**
- The electric field is a measure of the force experienced by a unit charge in space. The electric field for a planar charge distribution is given by the above formula.
2. **Surface Charge Density (\( \sigma \))**
- \( \sigma \) represents the amount of charge per unit area on the surface.
3. **Permittivity of Free Space (\( \epsilon_0 \))**
- \( \epsilon_0 \) is a constant that relates the units of electric charge to mechanical quantities such as force and energy (approximately \( 8.85 \times 10^{-12} \, \text{F/m} \)).
4. **Distance from the Charge (\( z \))**
- \( z \) denotes how far the point of interest is from the charged surface.
5. **R**
- \( R \) is a given constant in the equation which might represent a characteristic length like the radius in the context of a circular symmetric field.
### Steps for Calculation:
1. **Identify given values:** Understand and substitute the values given in the problem for \( \sigma \), \( \epsilon_0 \), \( z \), and \( R \).
2. **Simplify the expression:** Break down the equation step-by-step to solve for \( E \).
3. **Calculate potential \( V \):** In some contexts, electric potential \( V \) can be found by integrating the electric field, but instructions specific to \( E \) lead towards finding the value directly from this expression.
### Practical Usage:
Such calculations are fundamental in capacitor design, electrostatics problems, and understanding electric fields around planar charge distributions found in many](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe167cdfc-6687-4357-a9c7-ca435187bf42%2Fb9f73bbc-2e81-4370-857e-bd58ddbf6183%2Fvvkjjol_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Calculating Electric Potential from Electric Field
To find the electric potential from the electric field, we can start by understanding the given equation. Here is the specific equation for the electric field \( E \):
\[ E = \frac{\sigma}{2\epsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right) \]
Where:
- \( \sigma \) is the surface charge density.
- \( \epsilon_0 \) is the permittivity of free space.
- \( z \) is the distance from the charge.
- \( R \) is a given constant (possibly radius).
### Explanation of the Formula
1. **Electric Field (\( E \))**
- The electric field is a measure of the force experienced by a unit charge in space. The electric field for a planar charge distribution is given by the above formula.
2. **Surface Charge Density (\( \sigma \))**
- \( \sigma \) represents the amount of charge per unit area on the surface.
3. **Permittivity of Free Space (\( \epsilon_0 \))**
- \( \epsilon_0 \) is a constant that relates the units of electric charge to mechanical quantities such as force and energy (approximately \( 8.85 \times 10^{-12} \, \text{F/m} \)).
4. **Distance from the Charge (\( z \))**
- \( z \) denotes how far the point of interest is from the charged surface.
5. **R**
- \( R \) is a given constant in the equation which might represent a characteristic length like the radius in the context of a circular symmetric field.
### Steps for Calculation:
1. **Identify given values:** Understand and substitute the values given in the problem for \( \sigma \), \( \epsilon_0 \), \( z \), and \( R \).
2. **Simplify the expression:** Break down the equation step-by-step to solve for \( E \).
3. **Calculate potential \( V \):** In some contexts, electric potential \( V \) can be found by integrating the electric field, but instructions specific to \( E \) lead towards finding the value directly from this expression.
### Practical Usage:
Such calculations are fundamental in capacitor design, electrostatics problems, and understanding electric fields around planar charge distributions found in many
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