Problem 38
Question
Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by 2.20 \(\mathrm{cm} .\) (a) If the surface charge density for each plate has magnitude 47.0 \(\mathrm{nC} / \mathrm{m}^{2}\) , what is the magnitude of \(\vec{\boldsymbol{E}}\) in the region between the plates? (b) What is the potential difference between the two plates? (c) If the separation between the plates is doubled while the surface charge density is kept constant at the value in part (a), what happens to the magnitude of the electric field and to the potential difference?
Step-by-Step Solution
Verified Answer
(a) The electric field magnitude is 5313.56 N/C.
(b) The potential difference is 116.90 V.
(c) The electric field remains the same, but potential difference doubles.
1Step 1: Understanding Electric Field Due to Charged Plates
The formula to calculate the electric field between two parallel plates is given by:\[ E = \frac{\sigma}{\varepsilon_0} \]where \( \sigma \) is the surface charge density, and \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \) is the permittivity of free space.
2Step 2: Calculate the Electric Field
Plug in the values to find the electric field:\[ E = \frac{47.0 \times 10^{-9}}{8.85 \times 10^{-12}} \approx 5313.56 \, \text{N/C} \]Thus, the magnitude of the electric field \( E \) is approximately 5313.56 N/C.
3Step 3: Formula for Potential Difference Between Plates
The potential difference \( V \) between the plates is given by:\[ V = E \cdot d \]where \( E \) is the electric field and \( d \) is the separation between the plates.
4Step 4: Calculate the Potential Difference
Given \( d = 2.20 \, \text{cm} = 0.022 \, \text{m} \), substitute the values to determine \( V \):\[ V = 5313.56 \times 0.022 \approx 116.90 \, \text{V} \]Hence, the potential difference \( V \) between the plates is approximately 116.90 V.
5Step 5: Impact of Doubling Separation on Electric Field and Potential Difference
When the separation \( d \) is doubled, the electric field \( E = \frac{\sigma}{\varepsilon_0} \) remains unchanged since it is independent of \( d \) in a uniform field. However, the potential difference \( V = E \cdot d \) will also double since \( d \) is doubled while \( E \) is constant.
Key Concepts
Surface Charge DensityPotential DifferencePermittivity of Free SpaceParallel PlatesSeparation Distance
Surface Charge Density
Surface charge density, represented by \( \sigma \), describes how much charge per unit area is present on a surface. It is measured in coulombs per square meter (C/m²). In the context of electric fields between parallel plates, knowing \( \sigma \) is crucial because it helps determine the strength of the electric field in the area between the plates. A higher surface charge density means more charge is concentrated on the plate, leading to a stronger electric field.
For two oppositely charged plates, the magnitude of the surface charge density is equal on both plates but with opposite signs, ensuring an even electric field distribution between them. In effect, the electric field produced is proportional to the surface charge density, following the equation \( E = \frac{\sigma}{\varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space.
For two oppositely charged plates, the magnitude of the surface charge density is equal on both plates but with opposite signs, ensuring an even electric field distribution between them. In effect, the electric field produced is proportional to the surface charge density, following the equation \( E = \frac{\sigma}{\varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space.
Potential Difference
Potential difference (or voltage), expressed in volts (V), refers to the energy difference per unit charge between two points. In a parallel plate configuration, the potential difference between the plates is determined by the electric field and the distance separating the plates.
Mathematically, the potential difference \( V \) is given by the equation \( V = E \cdot d \), where \( E \) is the electric field strength and \( d \) is the separation distance between the plates.
This relationship implies that for a constant electric field, increasing the distance between the plates increases the potential difference linearly. Conversely, if the field strength increases and the distance remains constant, the potential difference also increases.
Mathematically, the potential difference \( V \) is given by the equation \( V = E \cdot d \), where \( E \) is the electric field strength and \( d \) is the separation distance between the plates.
This relationship implies that for a constant electric field, increasing the distance between the plates increases the potential difference linearly. Conversely, if the field strength increases and the distance remains constant, the potential difference also increases.
Permittivity of Free Space
The permittivity of free space, denoted by \( \varepsilon_0 \), is a constant that characterizes the ability of the vacuum (or free space) to permit electric field lines. This constant is crucial in electrostatics as it appears in the denominator of the equation for electric fields due to surface charge, \( E = \frac{\sigma}{\varepsilon_0} \).
The value of \( \varepsilon_0 \) is approximately \( 8.85 \times 10^{-12} \, \text{C}^{2}/\text{N} \cdot \text{m}^2 \).
This small value reflects that free space does not permit electric field lines to flow through it easily, and it fundamentally influences how electric fields behave in vacuum and other mediums. Understanding \( \varepsilon_0 \) is essential for predicting how charged objects interact over distances.
The value of \( \varepsilon_0 \) is approximately \( 8.85 \times 10^{-12} \, \text{C}^{2}/\text{N} \cdot \text{m}^2 \).
This small value reflects that free space does not permit electric field lines to flow through it easily, and it fundamentally influences how electric fields behave in vacuum and other mediums. Understanding \( \varepsilon_0 \) is essential for predicting how charged objects interact over distances.
Parallel Plates
Parallel plates are used in physics and engineering to create uniform electric fields. In this setup, two large, flat, conducting plates face each other, and each plate carries an equal but opposite charge.
This configuration ensures that the electric field between the plates is uniform, meaning it has the same magnitude and direction at every point between the plates, making calculations relatively straightforward. The strength of this electric field is determined by the surface charge density \( \sigma \) and the permittivity of free space \( \varepsilon_0 \), as expressed by the formula \( E = \frac{\sigma}{\varepsilon_0} \).
Parallel plate arrangements are essential in various applications, such as capacitors, which store electrical energy by maintaining a potential difference between the plates.
This configuration ensures that the electric field between the plates is uniform, meaning it has the same magnitude and direction at every point between the plates, making calculations relatively straightforward. The strength of this electric field is determined by the surface charge density \( \sigma \) and the permittivity of free space \( \varepsilon_0 \), as expressed by the formula \( E = \frac{\sigma}{\varepsilon_0} \).
Parallel plate arrangements are essential in various applications, such as capacitors, which store electrical energy by maintaining a potential difference between the plates.
Separation Distance
Separation distance, often denoted as \( d \), is the distance between two parallel plates. This measure plays a significant role in determining the potential difference between the plates.
Even though the electric field \( E \) between the plates is independent of \( d \), as shown by the equation \( E = \frac{\sigma}{\varepsilon_0} \), the potential difference \( V \) is directly proportional to \( d \). Therefore, increasing the distance increases the potential difference, assuming the electric field remains constant.
This principle highlights the importance of the separation distance in designing devices like capacitors, where control over voltage is critical. However, doubling \( d \) without changing \( \sigma \) keeps the field constant while doubling the potential difference, illustrating how geometric adjustments affect electrical properties.
Even though the electric field \( E \) between the plates is independent of \( d \), as shown by the equation \( E = \frac{\sigma}{\varepsilon_0} \), the potential difference \( V \) is directly proportional to \( d \). Therefore, increasing the distance increases the potential difference, assuming the electric field remains constant.
This principle highlights the importance of the separation distance in designing devices like capacitors, where control over voltage is critical. However, doubling \( d \) without changing \( \sigma \) keeps the field constant while doubling the potential difference, illustrating how geometric adjustments affect electrical properties.
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