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The concept related to solve this problem is the gauss’s law. First, calculate the magnitude of the electric field at a distance from the center of the ball. Later, calculate the magnitude of the electric field at a distance from the center of the ball. Finally, explain the electric field due to the charged ball throughout all of space and identify which of statement about the electric field is true.
Gauss’s Law: The flux of the net electric field through a closed surface equals the net charge enclosed by the surface divided by. Mathematically, the statement can be written as,
Here, is the electric field, is the infinitesimal surface element, q is the charge enclosed in the sphere, and is the permittivity of vacuum.
The expression for the electric field is,
Here, r is the radius of the sphere.
The charge enclosed in the sphere is written as:
Here, is the charge density of the sphere.
Part-A
Calculate the magnitude of the electric field at a distance from the center of the ball.
The expression for the gauss law is,
The area of the sphere is .
Substitute for in the above equation.
Here, q is the charge enclosed in the sphere, r is the radius of the sphere, and is the permittivity of free space.
The volume density of the sphere is written as:
The volume of the sphere is .
Rewrite the above equation as follows:
Substitute q value in the gauss law equation. The equation of electric field is written as:
Part-B
Calculate the charge enclosed in the sphere by using gauss law.
The charge enclosed in the sphere is written as:
Now, the equation for the electric field is written as:
Part-C
From the above two equations, when the radius r is zero then the electric field will be zero. At infinite distance, the electric field is zero. Maximum electric field is produced at a distance of . Hence, the correct options are 1, 3, 5.
Ans: Part-AThe magnitude of the electric field at a distance from the center of the ball is .
Part-BThe magnitude of the electric field at a distance from the center of the ball is
Part-CThe following statements are true. , , and the maximum electric field occurs when .
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