If the charge density of a spherical charge distribution increases linearly with radius, how does the electric field vary inside the sphere?

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Q: If the charge density of a spherical charge distribution increases linearly with radius, how does the electric field vary inside the sphere?

Solution: The electric field inside a non-uniform charge distribution varies quadratically with radius due to the increasing charge density.

Steps: 7

Step 1: Understand that charge density is how much charge is in a certain volume. In this case, it increases as you move away from the center of the sphere.
Step 2: Recognize that the electric field inside a sphere is affected by the charge distribution within it.
Step 3: Since the charge density increases linearly with radius, it means that as you go from the center to the surface, there is more charge in the outer layers.
Step 4: Use Gauss's Law, which relates the electric field to the charge enclosed within a certain radius.
Step 5: Calculate the total charge enclosed within a radius 'r' inside the sphere, which will involve integrating the charge density over the volume.
Step 6: Determine how the electric field depends on the total charge enclosed. Since the charge density increases with radius, the total charge will increase more than linearly.
Step 7: Conclude that the electric field inside the sphere will vary quadratically with radius because the total charge increases faster than the radius.