A slab of uniform nonconductive material is perpendicular to the z-axis, centere

Question

A slab of uniform nonconductive material is perpendicular to the z-axis, centered on the origin, has a total thickness of 40 mm, and is infinite along the other two axes. It has a uniform charge density of -108 pC/m^3. It contains a perfectly spherical bubble with no charge or material in it. The bubble is also centered on the origin n's has a radius of 10 mm. Find expressions for the electric field vector at all points on the positive z-axis. Use 9x10^-12 C^2/N•m^2 for the permittivity in all regions.
Please show all steps by using only variables first and then substituting in their given values. A slab of uniform nonconductive material is perpendicular to the z-axis, centered on the origin, has a total thickness of 40 mm, and is infinite along the other two axes. It has a uniform charge density of -108 pC/m^3. It contains a perfectly spherical bubble with no charge or material in it. The bubble is also centered on the origin n's has a radius of 10 mm. Find expressions for the electric field vector at all points on the positive z-axis. Use 9x10^-12 C^2/N•m^2 for the permittivity in all regions.
Please show all steps by using only variables first and then substituting in their given values.
Please show all steps by using only variables first and then substituting in their given values.

Explanation / Answer

For z> 20mm,

Formula for electric field due to sheet = sigma/2eo

E = Esheet+Ehole= - 108e-12*0.040/(2*9e-12) + 1/(4pi*9e-12)*108e-12*4/3*pi*0.010^3 /z

E = - 0.24 + 4*10^-6/z N/C where z is in m.

For z <10mm,

E = - 0.24*z/0.020 + rho z /3eo

= - 12z + 108e-12*z/(3*9e-12)

= - 12z + 4z

= - 8z N/C wherez is in m

For 10 mm <z< 20mm

= - 12z + 4*10^-6/z N/C where z is in m

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