Electric Field Outside a Solid Ball with Variable Charge Density

The problem states that we have a solid ball with radius R and a charge density that varies with the distance from the center of the ball. Let's analyze the electric field outside the ball using Gauss's law and the concept of charge enclosed.

1. Applying Gauss's Law:
Gauss's law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀). We can use this law to find the electric field outside the solid ball.

2. Gaussian Surface:
To apply Gauss's law, we need to choose an appropriate Gaussian surface. Since we are interested in the electric field outside the ball, we can select a spherical Gaussian surface with a radius greater than R.

3. Charge Enclosed:
The charge enclosed within our Gaussian surface is the total charge of the ball. To find this, we integrate the charge density ρ over the volume of the ball.

4. Electric Flux:
The electric flux through the Gaussian surface is given by the product of the electric field (E) and the surface area (A) of the Gaussian surface.

5. Applying Gauss's Law:
Using Gauss's law, we can equate the electric flux to the charge enclosed divided by ε₀. This gives us the equation:

Electric Flux = (Charge Enclosed) / ε₀

6. Electric Field Outside the Ball:
Since the electric field is radial and points outward, the electric flux is equal to E times the area of the Gaussian surface, which is 4πr² (where r is the distance from the center of the ball).

Therefore, we can rewrite the equation as:

E * 4πr² = (Charge Enclosed) / ε₀

7. Deriving the Electric Field:
To find the electric field outside the ball, we need to solve the above equation for E. Since the charge enclosed is obtained by integrating the charge density ρ over the volume of the ball, we substitute the expression for the charge enclosed into the equation.

E * 4πr² = (∫ρ dV) / ε₀

8. Charge Density Expression:
According to the problem, the charge density ρ is given by ρ₀(1 - r/R), where ρ₀ is a constant. By substituting this expression into the equation, we can solve for the electric field.

9. Solving for Electric Field:
After performing the integration and simplifications, the electric field outside the ball is found to be:

E = (ρ₀R³) / (3ε₀r²)

This expression shows that the electric field outside the solid ball is inversely proportional to the square of the distance from the center (1/r²). It also depends on the charge density ρ₀ and the radius R of the ball.

Conclusion:
In conclusion, the electric field outside a solid ball with variable charge density can be determined by applying Gauss's law and considering a Gaussian surface outside the ball. By integrating the charge density over the volume of the ball, the electric field is found to be (ρ₀R³) / (3ε₀r²).