A nonconducting sphere is made of two layers. The innermost section has a radius of 6.0 cm and a uniform charge density of −5.0C/m3. The outer layer has a uniform charge density of +8.0C/m3 and extends from an inner radius of 6.0 cm to an outer radius of 12.0 cm.

A) Determine the electric field for 0
B) Determine the electric field for 6.0cm
C) Determine the electric field for 12.0cm

a) E =0, b)   E = 1,129 10¹⁰ N / C , c)    E = 3.33 10¹⁰ N / C

Explanation:

To solve this exercise we can use Gauss's law

        Ф = ∫ E. dA = / ε₀

Where we must define a Gaussian surface that is this case is a sphere; the electric field lines are radial and parallel to the radii of the spheres, so the scalar product is reduced to the algebraic product.

           E A = /ε₀

The area of a sphere is

          A = 4π r²

         E = q_{int} / 4πε₀ r²

         k = 1 / 4πε₀

         E = k q_{int} / r²

To find the charge inside the surface we can use the concept of density

        ρ = q_{int} / V ’

         q_{int} = ρ V ’

         V ’= 4/3 π r’³

Where V ’is the volume of the sphere inside the Gaussian surface

 Let's apply this expression to our problem

a) The electric field in center r = 0

     Since there is no charge inside, the field must be zero

          E = 0

b) for the radius of r = 6.0 cm

In this case the charge inside corresponds to the inner sphere

        q_{int} = 5.0  4/3 π 0.06³

         q_{int} = 4.52 10⁻³ C

        E = 8.99 10⁹  4.52 10⁻³ / 0.06²

         E = 1,129 10¹⁰ N / C

c) The electric field for r = 12 cm = 0.12 m

In this case the two spheres have the charge inside the Gaussian surface, for which we must calculate the net charge.

     The charge of the inner sphere is q₁ = - 4.52 10⁻³ C

The charge for the outermost sphere is

       q₂ =  ρ 4/3 π r₂³

       q₂ = 8.0 4/3 π 0.12³

       q₂ = 5.79 10⁻² C

The net charge is

     q_{int} = q₁ + q₂

     q_{int} = -4.52 10⁻³ + 5.79 10⁻²

     q_{int} = 0.05338 C

The electric field is

        E = 8.99 10⁹ 0.05338 / 0.12²

        E = 3.33 10¹⁰ N / C

the answer is b ) square root of 5