What is the formula for the Gauss law of a sphere?
2 Answers
Gauss's law states that the electric flux \(\Phi_E\) through a closed surface is proportional to the charge \(Q_{\text{enc}}\) enclosed by that surface. For a spherical surface, the formula can be expressed as:
\[
\Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
\]
Where:
- \(\Phi_E\) is the electric flux through the surface.
- \(\mathbf{E}\) is the electric field.
- \(d\mathbf{A}\) is the differential area vector on the surface.
- \(Q_{\text{enc}}\) is the total charge enclosed within the sphere.
- \(\varepsilon_0\) is the permittivity of free space.
For a uniformly charged sphere, the electric field \(E\) at a distance \(r\) from the center (where \(r\) is greater than the radius \(R\) of the sphere) is given by:
\[
E = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r^2}
\]
This can be applied when determining the electric field outside a charged sphere using Gauss's law.
\[
\Phi_E = \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
\]
Where:
- \(\Phi_E\) is the electric flux through the surface.
- \(\mathbf{E}\) is the electric field.
- \(d\mathbf{A}\) is the differential area vector on the surface.
- \(Q_{\text{enc}}\) is the total charge enclosed within the sphere.
- \(\varepsilon_0\) is the permittivity of free space.
For a uniformly charged sphere, the electric field \(E\) at a distance \(r\) from the center (where \(r\) is greater than the radius \(R\) of the sphere) is given by:
\[
E = \frac{1}{4\pi \varepsilon_0} \frac{Q}{r^2}
\]
This can be applied when determining the electric field outside a charged sphere using Gauss's law.
Gauss's Law is a fundamental principle in electromagnetism that relates the electric field within a closed surface to the charge enclosed by that surface. When applied to a spherical symmetry, Gauss's Law can be expressed in a simplified form. Here's how it works for a sphere:
### Gauss's Law
Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it is given by:
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where:
- \( \Phi_E \) is the electric flux through the surface.
- \( Q_{\text{enc}} \) is the total charge enclosed by the surface.
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.854 \times 10^{-12} \, \text{F/m} \) (farads per meter).
### Applying Gauss's Law to a Sphere
For a spherical surface, we often use Gauss's Law to find the electric field created by a spherically symmetric charge distribution. Here's how you apply it:
1. **Choose a Gaussian Surface**: For a sphere, the Gaussian surface is also a sphere centered on the charge distribution. The choice of a spherical Gaussian surface simplifies the calculations due to symmetry.
2. **Calculate Electric Flux**: The electric flux \( \Phi_E \) through a spherical surface is given by the integral of the electric field \( \mathbf{E} \) over the surface area \( A \):
\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} \]
Since the electric field \( \mathbf{E} \) is radial and has the same magnitude at every point on the spherical surface, and the surface area \( A \) of a sphere is \( 4 \pi r^2 \), where \( r \) is the radius of the sphere, the flux simplifies to:
\[ \Phi_E = E \cdot 4 \pi r^2 \]
3. **Relate Flux to Charge**: By Gauss's Law, this flux is also equal to the charge enclosed divided by \( \epsilon_0 \):
\[ E \cdot 4 \pi r^2 = \frac{Q_{\text{enc}}}{\epsilon_0} \]
4. **Solve for the Electric Field**: Solving for the electric field \( E \), you get:
\[ E = \frac{Q_{\text{enc}}}{4 \pi \epsilon_0 r^2} \]
### Summary
For a sphere with a spherically symmetric charge distribution, Gauss's Law helps us determine that the electric field outside the sphere is given by:
\[ E = \frac{Q_{\text{enc}}}{4 \pi \epsilon_0 r^2} \]
where \( r \) is the distance from the center of the sphere to the point where the electric field is being calculated. Inside a conducting sphere with charge \( Q_{\text{enc}} \), the electric field is zero if the charge is distributed on the surface, while inside a non-conducting sphere with uniform charge density, it would vary differently and requires integration over the volume.
### Gauss's Law
Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it is given by:
\[ \Phi_E = \frac{Q_{\text{enc}}}{\epsilon_0} \]
where:
- \( \Phi_E \) is the electric flux through the surface.
- \( Q_{\text{enc}} \) is the total charge enclosed by the surface.
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.854 \times 10^{-12} \, \text{F/m} \) (farads per meter).
### Applying Gauss's Law to a Sphere
For a spherical surface, we often use Gauss's Law to find the electric field created by a spherically symmetric charge distribution. Here's how you apply it:
1. **Choose a Gaussian Surface**: For a sphere, the Gaussian surface is also a sphere centered on the charge distribution. The choice of a spherical Gaussian surface simplifies the calculations due to symmetry.
2. **Calculate Electric Flux**: The electric flux \( \Phi_E \) through a spherical surface is given by the integral of the electric field \( \mathbf{E} \) over the surface area \( A \):
\[ \Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{A} \]
Since the electric field \( \mathbf{E} \) is radial and has the same magnitude at every point on the spherical surface, and the surface area \( A \) of a sphere is \( 4 \pi r^2 \), where \( r \) is the radius of the sphere, the flux simplifies to:
\[ \Phi_E = E \cdot 4 \pi r^2 \]
3. **Relate Flux to Charge**: By Gauss's Law, this flux is also equal to the charge enclosed divided by \( \epsilon_0 \):
\[ E \cdot 4 \pi r^2 = \frac{Q_{\text{enc}}}{\epsilon_0} \]
4. **Solve for the Electric Field**: Solving for the electric field \( E \), you get:
\[ E = \frac{Q_{\text{enc}}}{4 \pi \epsilon_0 r^2} \]
### Summary
For a sphere with a spherically symmetric charge distribution, Gauss's Law helps us determine that the electric field outside the sphere is given by:
\[ E = \frac{Q_{\text{enc}}}{4 \pi \epsilon_0 r^2} \]
where \( r \) is the distance from the center of the sphere to the point where the electric field is being calculated. Inside a conducting sphere with charge \( Q_{\text{enc}} \), the electric field is zero if the charge is distributed on the surface, while inside a non-conducting sphere with uniform charge density, it would vary differently and requires integration over the volume.