A positively charged solid sphere of radius 100 mm has a uniform volume charge density of 250 nC/m^2. Determine the electric field (a) 20 mm, (b) 90 mm, and (c) 110 mm from the center of the sphere.

To determine the electric field at different distances from the center of a positively charged solid sphere, you can use the equation for the electric field due to a charged sphere. The equation is given by:

E = (k * Q) / r^2

where E is the electric field, k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2), Q is the total charge of the sphere, and r is the distance from the center of the sphere.

In this case, the sphere has a uniform volume charge density of 250 nC/m^2, which means that every square meter of the surface of the sphere has a charge of 250 nC. To determine the total charge of the sphere, we need to calculate the surface area of the sphere using its radius.

The surface area of a sphere is given by:

A = 4πr^2

where A is the surface area and r is the radius. Plug in the radius of the sphere (100 mm = 0.1 m) into the equation to find the surface area:

A = 4π(0.1)^2
A = 0.04π m^2

Next, we can calculate the total charge Q by multiplying the surface area A by the charge density:

Q = A * ρ
Q = 0.04π * 250 nC/m^2
Q = 100π nC

Now that we have the total charge of the sphere, we can calculate the electric field at different distances:

(a) At 20 mm (0.02 m) from the center of the sphere:
E = (k * Q) / r^2
E = (9 x 10^9 Nm^2/C^2 * 100π nC) / (0.02)^2
E = 4.5π x 10^13 N/C

(b) At 90 mm (0.09 m) from the center of the sphere:
E = (k * Q) / r^2
E = (9 x 10^9 Nm^2/C^2 * 100π nC) / (0.09)^2
E = 10π x 10^12 N/C

(c) At 110 mm (0.11 m) from the center of the sphere:
E = (k * Q) / r^2
E = (9 x 10^9 Nm^2/C^2 * 100π nC) / (0.11)^2
E = 5.6π x 10^12 N/C

Therefore, the electric field at (a) 20 mm from the center is approximately 4.5π x 10^13 N/C, (b) 90 mm from the center is approximately 10π x 10^12 N/C, and (c) 110 mm from the center is approximately 5.6π x 10^12 N/C.