Consider a spherical charge distribution of radius R with a uniform charge density ρ. Using Gauss'...

Consider a spherical charge distribution of radius R with a uniform charge density ρ.
Using Gauss' Law find the electric field at distance r from the axis where r < R.

Solutions

Expert Solution

First, consider a Gaussian sphere of radius inside the sphere of radius

Since a uniform charge density of is given, the final equation will be independent of the radius of the sphere.

Gauss's Law states that the electric flux through a symmetrical surface is equal to the enclosed charge divided by the absolute permittivity of space.

Here, to calculate , we can just use the density of charge times the volume of the charge distribution enclosed by the surface.

Evaluating , we get

Therefore, by Gauss's Law

We can see here that the electric field depends on the size of the Gaussian Surface taken, and is independent of the size of the charge distribution, provided

genius_generous answered 1 year ago

Next > < Previous

Related Solutions

1) An insulating sphere with radius R has a uniform positive volume charge density of ρ....

1) An insulating sphere with radius R has a uniform positive volume charge density of ρ. A solid metallic shell with inner radius R and outer radius 2R has zero total charge. [Express your answers for parts (a-d) using ρ, R, and constants] (a) What is the magnitude of the electric field at a distance ? = 3? away from the center? (b) Assuming the potential at infinity is 0. What is the potential at the outer surface (? =...

A sphere of radius R has a radius dependent charge density ρ = B · r3...

A sphere of radius R has a radius dependent charge density ρ = B · r3 in terms of R and B. Calculate the potential as a function of r from the center of the sphere.

Consider a spherical shell with radius R and surface charge density σ. By integrating the electric...

Consider a spherical shell with radius R and surface charge density σ. By integrating the electric field, find the potential outside and inside the shell. You should find that the potential is constant inside the shell. Why?

A spherical shell of radius a has a uniform surface charge density σ and rotates with...

A spherical shell of radius a has a uniform surface charge density σ and rotates with a constant angular velocity ω in relation to an axis that passes through its center. In this situation, determine the magnetic dipole moment μ of the spherical shell.

A long isolating cylinder with radius R and a charge density ρ(s) = 3λ πR3 (R...

A long isolating cylinder with radius R and a charge density ρ(s) = 3λ πR3 (R − s) for s ≤ R , 0 for s > R , where λ is a fixed positive line charge density (with units C/m) and s denotes the distance from the center of the cylinder. (a) Explain why the electric field is only a function of s. What is the direction of the electric field? (b) Use Gauss’ law to derive the magnitude...

The figure shows a spherical shell with uniform volume charge density ρ = 2.18 nC/m3, inner...

The figure shows a spherical shell with uniform volume charge density ρ = 2.18 nC/m3, inner radius a = 9.30 cm, and outer radius b = 2.6a. What is the magnitude of the electric field at radial distances (a) r = 0; (b) r = a/2.00, (c) r = a, (d) r = 1.50a, (e) r = b, and (f) r = 3.00b?

The figure shows a spherical shell with uniform volume charge density ρ = 1.88 nC/m3, inner...

The figure shows a spherical shell with uniform volume charge density ρ = 1.88 nC/m3, inner radius a = 9.70 cm, and outer radius b = 3.4a. What is the magnitude of the electric field at radial distances (a) r = 0; (b) r = a/2.00, (c) r = a, (d) r = 1.50a, (e) r = b, and (f) r = 3.00b?

An insulating sphere of radius a has charge density ρ(r) = ρ0r2, where ρ0 is a...

An insulating sphere of radius a has charge density ρ(r) = ρ0r2, where ρ0 is a constant with appropriate units. The total charge on the sphere is -3q. Concentric with the insulating sphere is a conducting spherical shell with inner radius b > a and outer radius The total charge on the shell is +2q. Determine (a) The magnitude of electric field at the following locations: (i) r < a; ii) a < r < b; (iii) b < r <...

A spherical cloud of negatively charged particles carries a charge density that is modeled by: ρ=ρ0(1−r/R)...

A spherical cloud of negatively charged particles carries a charge density that is modeled by: ρ=ρ0(1−r/R) Where ρ0 is the central density (which is 1.2 nC·m-3), R is the radius of the cloud (which is 2.9 m) and r is the distance from the centre of the cloud. a) Determine the electric field at 1.2 m from the centre of the cloud: __________ N·C-1 b) Determine the electric field at 4.1 m from the centre of the cloud: __________ N·C-1

A non conducting sphere of radius R and uniform volume charge density is rotating with angular...

A non conducting sphere of radius R and uniform volume charge density is rotating with angular velocity, Omega. Assuming the center of the sphere is at the origin of the coordinate system, a) what is the magnitude and direction of the resulting magnetic field on the z axis for any arbitrary z distance away from the origin when z > R? b) same question as part a) but for z < R? Omega of the rotating sphere on the extra...

Subjects