What is the relationship between surface density and electric intensity?
2 Answers
The relationship between surface charge density (\(\sigma\)) and electric field intensity (\(E\)) is established by **Gauss's law** in electrostatics.
### Surface Charge Density (\(\sigma\)):
Surface charge density represents the amount of charge per unit area on a surface, and is given by:
\[
\sigma = \frac{Q}{A}
\]
Where:
- \(Q\) is the total charge on the surface,
- \(A\) is the area of the surface.
### Electric Field Intensity (\(E\)):
Electric field intensity is a measure of the force per unit charge experienced by a test charge placed in the field.
### Relationship Between Surface Charge Density and Electric Field:
For a uniformly charged infinite plane, Gauss's law provides a direct relationship between the surface charge density (\(\sigma\)) and the electric field intensity (\(E\)) near the surface:
\[
E = \frac{\sigma}{\varepsilon_0}
\]
Where:
- \(E\) is the electric field intensity near the surface (in a direction perpendicular to the surface),
- \(\sigma\) is the surface charge density,
- \(\varepsilon_0\) is the permittivity of free space (\(8.854 \times 10^{-12} \, \text{F/m}\)).
### Explanation:
- The electric field intensity \(E\) is **directly proportional** to the surface charge density \(\sigma\). This means that as the surface charge density increases, the electric field intensity increases linearly.
- This relationship is derived using Gauss’s law, where the electric flux through a Gaussian surface is proportional to the enclosed charge.
### Important Assumptions:
- The plane is infinitely large or the charge distribution is uniform.
- The field is perpendicular to the surface and symmetric near the plane.
This formula applies to a variety of configurations such as conducting surfaces or planar charge distributions.
### Surface Charge Density (\(\sigma\)):
Surface charge density represents the amount of charge per unit area on a surface, and is given by:
\[
\sigma = \frac{Q}{A}
\]
Where:
- \(Q\) is the total charge on the surface,
- \(A\) is the area of the surface.
### Electric Field Intensity (\(E\)):
Electric field intensity is a measure of the force per unit charge experienced by a test charge placed in the field.
### Relationship Between Surface Charge Density and Electric Field:
For a uniformly charged infinite plane, Gauss's law provides a direct relationship between the surface charge density (\(\sigma\)) and the electric field intensity (\(E\)) near the surface:
\[
E = \frac{\sigma}{\varepsilon_0}
\]
Where:
- \(E\) is the electric field intensity near the surface (in a direction perpendicular to the surface),
- \(\sigma\) is the surface charge density,
- \(\varepsilon_0\) is the permittivity of free space (\(8.854 \times 10^{-12} \, \text{F/m}\)).
### Explanation:
- The electric field intensity \(E\) is **directly proportional** to the surface charge density \(\sigma\). This means that as the surface charge density increases, the electric field intensity increases linearly.
- This relationship is derived using Gauss’s law, where the electric flux through a Gaussian surface is proportional to the enclosed charge.
### Important Assumptions:
- The plane is infinitely large or the charge distribution is uniform.
- The field is perpendicular to the surface and symmetric near the plane.
This formula applies to a variety of configurations such as conducting surfaces or planar charge distributions.
The relationship between surface density and electric intensity is fundamental in electrostatics and can be understood through Gauss's Law. Here’s a detailed explanation:
### Surface Charge Density
Surface charge density (\(\sigma\)) is defined as the amount of electric charge per unit area on a surface. It is measured in coulombs per square meter (C/m²).
\[ \sigma = \frac{Q}{A} \]
where:
- \( Q \) is the total charge.
- \( A \) is the area over which the charge is distributed.
### Electric Intensity (Electric Field)
Electric intensity, or electric field (\(E\)), is a measure of the force per unit charge experienced by a positive test charge placed in the field. It is measured in volts per meter (V/m).
### Relationship via Gauss's Law
Gauss's Law relates the electric field to the surface charge density. For a uniformly charged infinite plane, Gauss's Law simplifies the relationship between the electric field and surface charge density:
\[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0} \]
where:
- \(\mathbf{E}\) is the electric field.
- \(d\mathbf{A}\) is the differential area vector.
- \(Q_{enc}\) is the charge enclosed by the Gaussian surface.
- \(\epsilon_0\) is the permittivity of free space (\(\approx 8.854 \times 10^{-12} \, \text{F/m}\)).
For an infinite plane with surface charge density \(\sigma\), the electric field is:
\[ E = \frac{\sigma}{2 \epsilon_0} \]
This formula shows that the electric field (\(E\)) due to an infinite plane of charge is directly proportional to the surface charge density (\(\sigma\)) and inversely proportional to the permittivity of free space (\(\epsilon_0\)).
### Key Points:
- The electric field due to an infinite plane of charge is constant in magnitude and is directed perpendicular to the surface of the plane.
- The factor of \(1/2\) in the formula arises because the plane contributes to the field on both sides of the surface.
In summary, the surface charge density \(\sigma\) and the electric field \(E\) are directly related by the formula \( E = \frac{\sigma}{2 \epsilon_0} \).
### Surface Charge Density
Surface charge density (\(\sigma\)) is defined as the amount of electric charge per unit area on a surface. It is measured in coulombs per square meter (C/m²).
\[ \sigma = \frac{Q}{A} \]
where:
- \( Q \) is the total charge.
- \( A \) is the area over which the charge is distributed.
### Electric Intensity (Electric Field)
Electric intensity, or electric field (\(E\)), is a measure of the force per unit charge experienced by a positive test charge placed in the field. It is measured in volts per meter (V/m).
### Relationship via Gauss's Law
Gauss's Law relates the electric field to the surface charge density. For a uniformly charged infinite plane, Gauss's Law simplifies the relationship between the electric field and surface charge density:
\[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0} \]
where:
- \(\mathbf{E}\) is the electric field.
- \(d\mathbf{A}\) is the differential area vector.
- \(Q_{enc}\) is the charge enclosed by the Gaussian surface.
- \(\epsilon_0\) is the permittivity of free space (\(\approx 8.854 \times 10^{-12} \, \text{F/m}\)).
For an infinite plane with surface charge density \(\sigma\), the electric field is:
\[ E = \frac{\sigma}{2 \epsilon_0} \]
This formula shows that the electric field (\(E\)) due to an infinite plane of charge is directly proportional to the surface charge density (\(\sigma\)) and inversely proportional to the permittivity of free space (\(\epsilon_0\)).
### Key Points:
- The electric field due to an infinite plane of charge is constant in magnitude and is directed perpendicular to the surface of the plane.
- The factor of \(1/2\) in the formula arises because the plane contributes to the field on both sides of the surface.
In summary, the surface charge density \(\sigma\) and the electric field \(E\) are directly related by the formula \( E = \frac{\sigma}{2 \epsilon_0} \).