The potential difference between the surface of a 3.0 -cm-diameter power line and a point \(1.0 \mathrm{m}\) distant is \(3.9 \mathrm{kV}\). Find the line charge density on the power line.

Electric potential difference, or voltage, is the amount of work needed to move a unit charge from one point to another in an electric field. Imagine you're pushing a ball uphill; the effort you exert to push the ball is similar to the work done by an external force to move a charge against the electric field.

The electric potential difference is crucial in understanding circuits and electric fields. It is typically measured in volts (V), and one volt is defined as the potential difference that will do one joule of work in moving one coulomb of charge. In our power line example, a potential difference of 3.9 kilovolts (or 3900 volts) implies that a considerable amount of work is needed to move charges through the 1-meter gap from the line's surface.

Permittivity of free space, denoted by \( \epsilon_0 \) is one of those fundamental constants in physics that play a vital role in electromagnetism. Think of it as a measure of how much resistance is encountered when forming an electric field in a vacuum. This value is crucial because it determines the strength of the electric field generated by a certain amount of charge in free space.

Coupled with the concept of the electric potential difference, permittivity of free space allows us to calculate the line charge density along the power line. It's like the ingredient in a recipe that determines how 'thick' or 'thin' our electric field soup turns out to be. The constant value of \( \epsilon_0 = 8.85 \times 10^-12 \, C^2/N \cdot m^2 \) helps us understand the relationship between electric potential, charge, and the resulting field in mathematical terms.

The way electric charge is distributed along an object greatly affects the electric field around it. In our textbook problem, the 'line charge density' gives us a picture of how charge is spread out along the length of the power line. This density, represented by the Greek letter \( \lambda \), indicates the amount of charge per unit length of the line.

Understanding charge distribution is key to analyzing and designing various electrical systems, such as capacitors, transmission lines, and even lightning rods. In the context of the exercise, once you know the line charge density, you can predict the potential difference at any point around the line and thus ensure the safe and effective operation of power systems. Calculating this density requires one to consider the potential difference, the permittivity of space, and the geometry of the charge distribution.