Sphere

Definition

A sphere is a three-dimensional geometric shape that is perfectly round, with every point on its surface equidistant from its center. Spheres are a fundamental concept in physics, particularly in the context of Gauss's Law, which describes the relationship between the electric flux through a closed surface and the total electric charge enclosed within that surface.

5 Must Know Facts For Your Next Test

1. The electric flux through a closed spherical surface is proportional to the total electric charge enclosed within that surface, as described by Gauss's Law.
2. The electric field outside a spherically symmetric charge distribution is radial and inversely proportional to the square of the distance from the center of the sphere.
3. The electric field inside a uniformly charged sphere is linearly proportional to the distance from the center of the sphere.
4. Gauss's Law can be used to easily calculate the electric field for spherically symmetric charge distributions, as the electric flux through any closed spherical surface is simply proportional to the enclosed charge.
5. Spherical symmetry is a common assumption in many physics problems, as it simplifies the calculations and allows for the application of Gauss's Law.

Review Questions

• Explain how the electric flux through a closed spherical surface is related to the total electric charge enclosed within that surface, according to Gauss's Law.
  • According to Gauss's Law, the electric flux \Phi_E through a closed spherical surface is proportional to the total electric charge Q enclosed within that surface. Specifically, the electric flux is given by the formula \Phi_E = \frac{Q}{\epsilon_0}, where \epsilon_0 is the permittivity of free space. This relationship holds true for any closed surface, but the spherical geometry simplifies the calculations, as the electric field outside a spherically symmetric charge distribution is radial and inversely proportional to the square of the distance from the center of the sphere.
• Describe the electric field inside and outside a uniformly charged sphere, and explain how Gauss's Law can be used to determine these fields.
  • For a uniformly charged sphere, Gauss's Law can be used to easily determine the electric field both inside and outside the sphere. Outside the sphere, the electric field is radial and inversely proportional to the square of the distance from the center of the sphere, E = \frac{Q}{4\pi\epsilon_0r^2}, where r is the distance from the center. Inside the sphere, the electric field is linearly proportional to the distance from the center, E = \frac{Q}{4\pi\epsilon_0R^3}r, where R is the radius of the sphere. This is because the electric flux through any closed spherical surface inside the sphere is proportional to the enclosed charge, allowing the electric field to be calculated directly from Gauss's Law.
• Analyze the importance of spherical symmetry in the application of Gauss's Law, and discuss how this simplifies the calculation of electric fields in physics problems.
  • Spherical symmetry is a crucial assumption in many physics problems, as it allows for the straightforward application of Gauss's Law to calculate electric fields. When a charge distribution has spherical symmetry, the electric field outside the distribution is radial and inversely proportional to the square of the distance from the center. This means that the electric flux through any closed spherical surface surrounding the charge distribution is simply proportional to the total enclosed charge, according to Gauss's Law. This simplifies the calculations compared to more complex charge distributions, as the electric field can be determined directly from the enclosed charge, without the need for detailed knowledge of the charge distribution itself. The spherical geometry also allows for the use of symmetry arguments to further simplify the problem, making spherically symmetric charge distributions a common and important assumption in physics.

Related terms

Electric Flux:

The electric flux is a measure of the total electric field passing through a given surface, and is a key concept in Gauss's Law.

Gaussian Surface:

A Gaussian surface is an imaginary closed surface used to apply Gauss's Law, often in the shape of a sphere, cube, or other geometric shape.

Electric Charge:

Electric charge is a fundamental property of matter that can be positive or negative, and is the source of the electric field that Gauss's Law describes.