A Spherical Region Of Radius A Has Total Charge Q. If The Charge Is Uniformly Distributed, Apply Gauss's Law To Find D Both Inside And Outside The Sphere.

Mar 10, 2025 by ADMIN 155 views

**A Spherical Region of Radius a has Total Charge Q: Applying Gauss's Law** **Introduction**

Gauss's law is a fundamental concept in physics that relates the distribution of electric charge to the resulting electric field. In this article, we will apply Gauss's law to a spherical region of radius a with total charge Q, uniformly distributed. We will derive the electric field both inside and outside the sphere.

Gauss's law states that the total electric flux through a closed surface is proportional to the charge enclosed within that surface. Mathematically, it is expressed as:

∮E * dA = Q / ε₀

where E is the electric field, dA is the area element of the surface, Q is the charge enclosed, and ε₀ is the electric constant (also known as the permittivity of free space).

Let's consider a spherical region of radius a with total charge Q, uniformly distributed. We will apply Gauss's law to this region to find the electric field both inside and outside the sphere.

Inside the Sphere

To find the electric field inside the sphere, we will use a Gaussian surface that is a sphere of radius r, where r < a. The electric field is constant and directed radially outward, so the electric flux through the surface is:

∮E * dA = E * 4πr²

The charge enclosed within the surface is:

Q = (4/3) * π * r³ * ρ

where ρ is the charge density.

Substituting these expressions into Gauss's law, we get:

E * 4πr² = (4/3) * π * r³ * ρ / ε₀

Simplifying and solving for E, we get:

E = ρr / 3ε₀

Outside the Sphere

To find the electric field outside the sphere, we will use a Gaussian surface that is a sphere of radius r, where r > a. The electric field is constant and directed radially outward, so the electric flux through the surface is:

∮E * dA = E * 4πr²

The charge enclosed within the surface is:

Q = (4/3) * π * a³ * ρ

Substituting these expressions into Gauss's law, we get:

E * 4πr² = (4/3) * π * a³ * ρ / ε₀

Simplifying and solving for E, we get:

E = Q / 4πε₀r²

In this article, we applied Gauss's law to a spherical region of radius a with total charge Q, uniformly distributed. We derived the electric field both inside and outside the sphere. The results show that the electric field inside the sphere is proportional to the radius, while the electric field outside the sphere is inversely proportional to the square of the distance from the center of the sphere.

Q: What is Gauss's law? A: Gauss's law is a fundamental concept in physics that relates the distribution of electric charge to the resulting electric field.

Q: What is the electric flux through a closed surface? A: The electric flux through a closed surface is proportional to the charge enclosed within that surface.

Q: How do you apply Gauss's law to a spherical region? A: To apply Gauss's law to a spherical region, you need to use a Gaussian surface that is a sphere of radius r, where r < a or r > a.

Q: What is the electric field inside a sphere with uniformly distributed charge? A: The electric field inside a sphere with uniformly distributed charge is proportional to the radius.

Q: What is the electric field outside a sphere with uniformly distributed charge? A: The electric field outside a sphere with uniformly distributed charge is inversely proportional to the square of the distance from the center of the sphere.

Q: What is the charge density? A: The charge density is the amount of charge per unit volume.

Q: What is the electric constant? A: The electric constant is also known as the permittivity of free space.

Q: What is the electric flux through a surface? A: The electric flux through a surface is the amount of electric field that passes through the surface.

Q: What is the electric field? A: The electric field is a vector field that describes the force that a charged particle would experience if placed in the field.

Q: What is the Gaussian surface? A: The Gaussian surface is a surface that is used to calculate the electric flux through a closed surface.

Q: What is the electric potential? A: The electric potential is the potential energy per unit charge at a given point in an electric field.

Q: What is the electric field strength? A: The electric field strength is the magnitude of the electric field at a given point.

Q: What is the electric flux density? A: The electric flux density is the amount of electric flux per unit area.

Q: What is the electric displacement field? A: The electric displacement field is a vector field that describes the amount of electric flux per unit area.

Q: What is the electric field intensity? A: The electric field intensity is the magnitude of the electric field at a given point.

Q: What is the electric field gradient? A: The electric field gradient is the rate of change of the electric field with respect to distance.

Q: What is the electric field distribution? A: The electric field distribution is the way in which the electric field is distributed in space.

Q: What is the electric field pattern? A: The electric field pattern is the way in which the electric field is distributed in space.

Q: What is the electric field configuration? A: The electric field configuration is the way in which the electric field is distributed in space.

Q: What is the electric field orientation? A: The electric field orientation is the direction in which the electric field is pointing.

Q: What is the electric field magnitude? A: The electric field magnitude is the magnitude of the electric field at a given point.

Q: What is the electric field direction? A: The electric field direction is the direction in which the electric field is pointing.