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The electric field is a crucial concept in electrostatics, representing the force per unit charge that would act on a small positive test charge in the vicinity of other charges. In this context, we have a uniformly distributed charge within a sphere. Using Gauss's Law, we can find the electric field inside the sphere.

The law helps us understand how charges behave within symmetrical configurations, like this sphere. For a distance from the center of the sphere where the radius is lesser than the sphere's radius, the electric field is given by:

• \( E(r) = \frac{q r}{4 \pi \varepsilon_0 R^3} \)

This indicates that the electric field inside the sphere increases linearly with the distance from the center, directly proportional to the radius \( r \).

The proportionality to \( r \) means that if you move further from the center inside this sphere, the electric field strength increases until you reach the surface of the sphere.

Potential difference is related to the work done in moving a charge from one point to another within an electric field. In this problem, we are interested in the electric potential inside a spherical charge distribution.

Electric potential \( V \) at a distance \( r \) inside a charged sphere can be calculated by integrating the electric field from a point at infinity to \( r \). This integral accounts for how the field changes as one moves inside the sphere:

• \( V(r) = \frac{q}{8 \pi \varepsilon_0 R^3} (3R^2 - r^2) \)

This expression shows that within the sphere, the potential decreases as we move from the center \( r = 0 \) to the edge \( r = R \).

Moreover, the potential difference between the edge and the center of the sphere is:

• \( \Delta V = V(R) - V(0) = -\frac{q}{4 \pi \varepsilon_0 R} \)

This negative sign indicates a loss in potential energy as we move towards the center of the sphere from its surface.

Gauss's Law is a pivotal principle in electrostatics, connecting the electric fields at a boundary with the charges within. It simplifies the calculation of the electric field when dealing with symmetric charge distributions, such as spheres.

In this context, Gauss's Law allows the calculation of electric fields inside the sphere by considering a virtual Gaussian surface within. The formula used arises from Gauss's Law and symmetry. It implies:

• The electric field inside a uniformly charged sphere varies linearly with the radius.
• Outside such a sphere, if the sphere is uniformly charged, the electric field is akin to that of a point charge located at the center.

Understanding how Gauss’s Law works is key because it helps reduce complex field calculations into manageable equations by leveraging the symmetry properties of the charge distribution. By applying Gauss’s Law, we gain a better grasp of how charges create electric fields in symmetric configurations.