Respuesta :
Answer:
E = Q / 4 π ε₀ (r-R_int) / (R_put³ -R_int³)
Explanation:
For this exercise we can use Gauss's law
Ф = ∫ E. dA = [tex]q_{int}[/tex] / ε₀
Where q is the charge inside the surface.
In this case the surface must be a sphere, the electric field lines and the radius of the sphere are parallel, so the scalar product is reduced to the algebraic product
∫ E dA = q_{int}/ ε₀
The area of a sphere is
A = 4π r²
- The electric field for a distance r < R_int
The charge inside is zero, so the electric field
E = 0 r <R_in t
- The field for a radio inside the shell
Let's use the concept of density
ρ = Q / V
q = ρ (4/3 π r³)
dq = ρ 4π r² dr
We substitute in the Gaussian equation
E ∫ dA = ρ 4π r² dr / ε₀
E 4π r² = ρ 4π/ε₀ r³ / 3
E = ρ / 3ε₀ r
We evaluate between the lower limit r = R_int, E = 0 and the upper limit r = r, E = E
E- 0 = ρ / 3ε₀ (r –R_int)
Density is
ρ = q / 4/3 π (R_out³ - R_int³)
Where R <r
E = Q / 4 π ε₀ (r-R_int) / (R_put³ -R_int³)
To answer that question we need to use the Gauss theorem of the flux of a vectorial field over a closed surface, in fact, this is the first of Maxwell´s equations for electromagnetism:
∫s) E×dA = dq/ε₀ (1) E and dA ( are vectors )
Solution is:
for Rin< r ≤ Rout E = Q×r /4×π×ε₀×(Rout³ - Rin³)
for r < Rin E = 0
We know:
- Every imaginary concentric shell with the spherical charge shell, is a gaussian surface ( that means all of them are equipotential surfaces E = constant)
- The charge is uniformly distributed over the volume of the spherical shell
- dq is a function of r (the distance from de center of the sphere )
- If ρ is the density of charge
dq = ρ×4×π×r²×dr ⇒ q = ×4×π×r³/3
Q = ρ×4×π×r³/3
By substitution in (1)
E × 4×π×r² = Q/ε₀ ⇒ E × 4×π×r² = ρ×4×π×r³/3×ε₀
E = ρ×r/3×ε₀ (2)
We need to express E as a function of Rout and Rin and Q
The density of charge
ρ = Q / Vs Vs = (4/3)×π×R³ ⇒ Vs = (4/3)×π×(Rout³ - Rin³)
ρ = Q /(4/3)×π×(Rout³ - Rin³)
And finally plugging the value of ρ in equation (2)
E = [Q× r/(4/3)×π×(Rout³ - Rin³)] /3×ε₀
E = Q×r /4×π×ε₀×(Rout³ - Rin³) for Rin< r ≤ Rout
For r < Rin the enclosed charge is 0 then E = 0
Note r = Rg in the attached file
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