Electric Field at a Distance of 3 cm from the Center of a Dielectric Sphere

Given:
Electric field at a distance of 20 cm from the center of the dielectric sphere = 100 V/m
Radius of the dielectric sphere = 10 cm

We need to find the electric field at a distance of 3 cm from the center of the sphere.

To solve this problem, we will use the concept of Gauss's Law and the formula for the electric field of a uniformly charged sphere.

1. Applying Gauss's Law:
Gauss's Law states that the flux of the electric field through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of the medium.

2. Electric Field of a Uniformly Charged Sphere:
The electric field of a uniformly charged sphere at a distance r from its center is given by the equation:
E = k * (Qenc / r^2)
where E is the electric field, k is the electrostatic constant, Qenc is the charge enclosed by the Gaussian surface, and r is the distance from the center of the sphere.

Now, let's calculate the electric field at a distance of 3 cm from the center of the dielectric sphere.

3. Calculating Charge Enclosed:
To calculate the charge enclosed, we need to know the charge distribution within the dielectric sphere. However, the given information does not provide any details about the charge distribution.

4. Assuming Uniform Charge Distribution:
Let's assume that the charge distribution within the dielectric sphere is uniform. This assumption allows us to consider the entire sphere as a point charge located at its center.

5. Electric Field Calculation:
Using the formula for the electric field of a uniformly charged sphere, we can calculate the electric field at a distance of 3 cm from the center of the sphere.

E = k * (Qenc / r^2)

Given:
Radius of the dielectric sphere = 10 cm
Electric field at a distance of 20 cm from the center = 100 V/m

Calculations:
Let's assume the charge enclosed by the Gaussian surface as Qenc.

At a distance of 20 cm from the center of the sphere, the electric field is given as 100 V/m. Using this information, we can calculate the charge enclosed by the Gaussian surface using the formula:

100 = k * (Qenc / 20^2)

Simplifying the equation:

Qenc = (100 * 20^2) / k

Now, we can calculate the electric field at a distance of 3 cm from the center of the sphere using the formula:

E = k * (Qenc / r^2)

E = k * (Qenc / 3^2)

Substituting the value of Qenc:

E = k * ((100 * 20^2) / k) / 3^2

Simplifying the equation:

E = (100 * 20^2) / (k * 3^2)

Finally, we can calculate the electric field at a distance of 3 cm from the center of the dielectric sphere using the above equation.

Note: The value of k is the electrostatic constant, which is approximately 9 x 10^9 Nm^2/C^2. By substituting the value of k in the equation, we can find the numerical value of the electric field.