A uniform line charge of \(2 \mu \mathrm{C} / \mathrm{m}\) is located on the \(z\) axis. Find \(\mathbf{E}\) in rectangular coordinates at \(P(1,2,3)\) if the charge exists from \((a)-\infty<\) \(z<\infty ;(b)-4 \leq z \leq 4\).

Understanding the concept of line charge density is crucial when calculating the electrical properties of extended charge distributions. Line charge density, often symbolized by \(\lambda\), measures the amount of electric charge per unit length along a line. It is typically expressed in coulombs per meter (C/m).

For instance, a uniform line charge density means that the charge is distributed evenly along the line, with the same amount of charge per segment of the line. In practical problems, we often deal with finite or infinite lines of charge, and line charge density becomes a central factor in determining the electric field these lines create at a point in space.

For example, if a line charge stretches to infinity in both directions, the symmetry simplifies the calculations and often leads to a zero net electric field in certain directions due to the cancellation of opposing field components. Conversely, a finite line charge, such as the one between -4 and 4 meters in our example, requires careful integration over the specified limits to determine the net electric field effect at a given point.

Coulomb's Law is foundational to understanding electric force between two point charges. It states that the electric force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it's expressed as \( F = k \frac{q_1 q_2}{r^2} \), where \(F\) is the force, \(q_1\) and \(q_2\) are the charges, \(r\) is the distance between them, and \(k\) is Coulomb's constant.

When dealing with extended charges, such as a line of charge, we apply Coulomb's law to infinitesimally small segments of the charge. Here, we introduce the line charge density \(\lambda\) to represent the charge per unit length. For an infinitesimal charge element \(\delta q\), Coulomb's law can be modified to \(\delta \mathbf{E} = \frac{1}{4\pi \epsilon_0}\frac{\delta q}{r^2} \hat{r}\), where \(\delta \mathbf{E}\) is the differential electric field created by \(\delta q\). The challenge is then to integrate these infinitesimal electric field contributions over the entire length of the charge distribution to acquire the total electric field.

Electromagnetic integration is the mathematical process we use to find the total electric field resulting from a continuous charge distribution. To compute the electric field at a specific point due to a line charge, we divide the line into infinitesimally small segments, each contributing a tiny electric field \(\delta \mathbf{E}\). We then sum up (integrate) all these contributions over the length of the charge distribution.

In our example, for the entire infinite line charge, we integrate from negative to positive infinity, essentially adding up the effects of all charges along the z-axis. However, for the finite line charge, we limit our integration from -4 to 4 meters along the z-axis. This process is more complex than the former, as it lacks the simplifying symmetry of an infinite line.

The actual integration is carried out by breaking down \(\delta \mathbf{E}\) into its components along the x, y, and z coordinates, and then performing a definite integral for each component within the specified limits. Electromagnetic integration allows us to move from discrete or differential charges to a full picture of the electric field in a given region, illustrating the strength and direction of the field across the space.