the region 0 < r < r1 is a charged insulating sphere with
uniform charge density _E, the region r1 < r < r2 contains free space, and the
region r2 < r < r3 is an insulating spherical shell with uniform charge density
��_E.
What is the electric field in the regions:
(a) 0 < r < r1 = 0
(b) r1 < r < r2
(c) r2 < r < r3
(d) r3 < r
(e) For which regions does the electric field have the same r dependence as the
field due to a point charge?
I need the answer to this question
To find the electric field in each region, we will use Gauss's law. Gauss's law relates the electric field to the enclosed charge within a closed surface.
Before we begin, it is important to note that the charge distribution is spherically symmetric in this problem. This means that the electric field will only depend on the radial distance from the center of the charge distribution.
(a) In the region 0 < r < r1, the electric field is zero. Inside the charged insulating sphere, the electric field is zero everywhere. This is because the charge is distributed uniformly throughout the entire sphere and cancels out the electric field at every point inside.
(b) In the region r1 < r < r2, we consider a Gaussian surface in the form of a sphere of radius r centered at the origin. The enclosed charge within this surface is the charge density multiplied by the volume of the region, which is given by (4/3)πr2^3 - (4/3)πr1^3. Applying Gauss's law, we can write:
∮ E · dA = (1/ε₀) ∫ ρ dV
Since the electric field is spherically symmetric, its magnitude is constant on the Gaussian surface. Therefore, the dot product E · dA becomes E * (4πr^2), where r is the radius of the Gaussian surface. The left-hand side becomes 4πr^2E, and the right-hand side becomes (1/ε₀) * ρ * (4/3)π(r2^3 - r1^3).
Setting the left-hand side equal to the right-hand side and solving for E, we find:
E = (ρ/3ε₀) * (r2^3 - r1^3) / r^2
(c) In the region r2 < r < r3, we can perform the same steps as in part (b) to find the electric field. The enclosed charge within the Gaussian surface is given by the charge density multiplied by the volume of the region, which is (4/3)π(r3^3 - r2^3). Applying Gauss's law, we get:
∮ E · dA = (1/ε₀) ∫ ρ dV
Again, since the electric field is spherically symmetric, the dot product E · dA becomes E * (4πr^2). The left-hand side simplifies to 4πr^2E, and the right-hand side becomes (1/ε₀) * ρ * (4/3)π(r3^3 - r2^3).
Setting the left-hand side equal to the right-hand side and solving for E, we find:
E = (ρ/3ε₀) * (r3^3 - r2^3) / r^2
(d) In the region r3 < r, the electric field is zero. Beyond the insulating spherical shell, the electric field is zero everywhere. This is because there is no charge present in this region.
(e) The electric field due to a point charge, Q, at a distance r from the charge is given by Coulomb's law:
E = kQ/r^2
Comparing this expression to the electric field expressions we derived in parts (a), (b), and (c), we can observe that the electric field in these regions does not have the same r dependence as the field due to a point charge. In these regions, the electric field depends on the difference in the volumes of the charge distribution (sphere or shell) rather than solely on the inverse square of the distance (r).