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Charge per unit length, often denoted by the symbol \( \lambda \), is a measure of how much electric charge is distributed along a line of charge. It's akin to density but instead of mass, we deal with charge. When a line of charge is uniform, the charge is evenly distributed along its length, meaning that each segment of the line carries the same amount of charge per unit of length.

For example, if we know the electric field intensity a certain distance from a line charge, we can find out the charge per unit length by rearranging the formula \( E = \frac{\lambda}{2\pi\varepsilon_0 r} \). The given values from the exercise can then be plugged in, allowing us to calculate \( \lambda \). This is a fundamental step in understanding the distribution of charge along the line and is crucial for solving various electrostatic problems.

Linear charge density, which is essentially the same as charge per unit length, is typically expressed in coulombs per meter (\(C/m\)). It's an idealized concept usually used for infinitely or very long conductors where we ignore the ends, assuming the charge distribution is not affected by edge effects.

In our example, we are dealing with a 'very long uniform line of charge,' which signifies that the linear charge density will remain constant no matter which segment of the line we examine. This allows us to use the uniform linear charge density to determine the total charge within any given length of the line. This concept underpins the relationship between charge distributions and the resultant electric fields.

The electric field intensity, denoted as \( E \), represents the force experienced by a unit positive charge in an electric field and is measured in newtons per coulomb (\(N/C\)). It is a vector quantity, which means it has both magnitude and direction, pointing away from positive charges and toward negative charges.

In our textbook exercise, we are specifically looking at the electric field intensity due to a line charge. Here, the formula that connects electric field intensity with linear charge density and distance from the line charge is fundamental. The intensity of the electric field diminishes as one moves further away from the line charge, and this inverse relationship with distance can be observed in the equation used to solve the problem. Understanding how to manipulate this equation is key to solving many problems in electrostatics.

Coulomb's constant, \( k \), is a proportionality factor that appears in Coulomb's law, which describes the force between two point charges. Its value is approximately \( 8.99 \times 10^9 \mathrm{~N\cdot m^2/C^2} \). However, in calculations involving electric fields and charge distributions, we more commonly use the electric constant \( \varepsilon_0 \), also known as the permittivity of free space.

In the context of a line charge, \( \varepsilon_0 \) is crucial as it appears in the denominator of the formula connecting the electric field intensity and the charge per unit length. The precise value of \( \varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~C^2/N\cdot m^2} \) is used in our calculation and ensures that the electric field is accurately described in standard units. It is important to understand the role of \( \varepsilon_0 \) to properly connect the concepts of electric fields with charge distributions.