A solid copper sphere of radius \(R=10 \mathrm{~cm}\) carries a charge of \(+5 \mathrm{nC}\). a) Give an algebraic expression for the electric field at all distances \(r

Gauss's Law is a fundamental principle describing the distribution of electric charge around a conductor and the resulting electric field. It states that the electric flux through a closed surface is directly proportional to the enclosed electric charge. The law can be mathematically expressed as \(\Phi_E = \frac{Q_{\text{enc}}}{\varepsilon_0} \) with \(\Phi_E\) representing the electric flux, \(Q_{\text{enc}}\) the enclosed charge, and \(\varepsilon_0\) the permittivity of free space.

For symmetrical charge distributions, like a uniformly charged sphere, Gauss's Law tremendously simplifies calculating the electric field. By choosing a Gaussian surface that matches the symmetry of the charge distribution, we can easily relate the enclosed charge to the electric field at every point on the surface. In our exercise concerning a solid copper sphere, applying Gauss's Law allows us to determine the electric field inside the sphere based on charge distribution.

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric field. Charges come in two types, positive and negative, commonly carried by protons and electrons, respectively. It is quantized, meaning it exists in discrete packets (like the charge of a single electron).

In our problem, we look at a solid copper sphere carrying a positive charge of \(+5\text{nC}\) (nanocoulombs). This charge influences the electric field and potential around and within the sphere. Understanding the behavior of electric charges, especially how they distribute in conductors (like copper) and insulators, is essential to grasp concepts like electric fields, potentials, and the forces between charged objects.

Electric potential is the potential energy per unit charge due to an electric field. It provides insight into the work done by or against the electric field when moving a charge within. Electric potential is a scalar quantity, meaning it doesn't have a direction, and it is expressed in volts (V).

The formula for electric potential \(V\) at a distance \(r\) is derived from integrating the electric field \(E\) over the distance from a reference point. In the case of our charged sphere, we find the potential at a point inside the sphere by integrating from the surface of the sphere to the point of interest, reflecting the work done in moving a unit charge inwards.

Charge distribution in solids, especially in conductors like the copper sphere in our exercise, is governed by the material's ability to allow charge movement. In a solid conductor at equilibrium, free charges move until they are uniformly distributed across the surface. This results in zero net electric field inside the conductor.

However, when discussing non-conducting solids, or dielectrics, the charge can be distributed throughout the volume of the material. As seen in our problem, the charge given to the copper sphere is thought to remain uniformly distributed within its volume, influencing the creation of an electric field that varies with distance from the center of the sphere.

Coulomb's Law is central to understanding electrostatic interactions between charged particles. According to the law, the force \(F\) between two point charges is directly proportional to the product of the magnitudes of charges \(q_1\) and \(q_2\), inversely proportional to the square of the distance \(r\) between them, and depends on the medium's permittivity \(\varepsilon\). Mathematically, Coulomb's Law is represented as \(F = k \frac{|q_1 q_2|}{r^2}\), where \(k\) is Coulomb's constant.

In the context of our electric field problem, outside the sphere we would apply Coulomb's Law to find the electric field generated by the sphere at a point distance \(r\) greater than its radius.