A solid sphere \(10 \mathrm{cm}\) in radius carries a \(40-\mu \mathrm{C}\) charge distributed uniformly throughout its volume. It's surrounded by a concentric shell \(20 \mathrm{cm}\) in radius, also uniformly charged with \(40 \mu \mathrm{C}\). Find the electric field (a) \(5.0 \mathrm{cm},\) (b) \(15 \mathrm{cm},\) and (c) \(30 \mathrm{cm}\) from the center.

Coulomb's law is a principle that quantifies the amount of force between two stationary, electrically charged particles. The formula for Coulomb's law is:
\[ F = K_e \frac{|q_1 \times q_2|}{r^2} \]
where \( F \) is the electric force between the charges, \( K_e \) is Coulomb's constant (approximately \( 8.9875 \times 10^9 \) N·m²/C²), \( q_1 \) and \( q_2 \) are the quantities of the charges, and \( r \) is the distance between the centers of the two charges. This law helps us understand how charged objects interact through an invisible electric field, which is especially useful when dealing with charge distributions like uniformly charged spheres.
In the context of the exercise, Coulomb's law is employed to calculate the electric field, which is essentially the force per unit charge experienced by a small positive test charge placed in the field.

Charge distribution refers to how electric charge is spatially arranged within an object. It can be uniform or non-uniform. In the exercise provided, we have a solid sphere and a concentric shell, both with charges that are uniformly distributed throughout their volumes.

When the charge distribution is uniform, the electric field within and around the object can be calculated using simplified assumptions. For example, inside a uniformly charged sphere, the electric field is directly proportional to the distance from the center, and it can be thought of as if the charge was concentrated in a smaller sphere with equivalent charge density, as shown in Step 1 of the solution.

Electrostatics is the study of electric charges at rest. The fundamental concept of electrostatics is that like charges repel, while unlike charges attract each other. An important aspect of electrostatics is the creation of an electric field around any charged object. This field exerts forces on other charges within the field's influence.

In our exercise, electrostatic principles help determine the behavior of the electric field at various distances from the charged sphere and shell. Inside a conductor at electrostatic equilibrium, the electric field is zero. This fact is implicit in determining that at the phase of the solution where we calculate the electric field at a point within the shell and beyond the solid sphere, only the charge of the solid sphere matters, as in Step 2.

A uniformly charged sphere is a common model in electrostatics that assumes charge is spread evenly across the volume of a sphere. Because of its symmetry, the physics simplifies and yields elegant solutions to otherwise complex problems. For a sphere of radius \( R \) and total charge \( Q \), when outside the sphere, the electric field is as if all the charge were concentrated at the center, given by the formula:
\[ E = \frac{K_e \times Q}{r^2} \]
And for points inside, it varies linearly with distance from the center, which can be derived from Gauss's law, another fundamental principle in electrostatics related to charge distribution.
In our exercise scenario, at the point within the solid sphere, the charge contributing to the electric field is less than the total charge of the sphere, proportionate to the volume, as described in Step 1.