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The electric field is a crucial concept when discussing charged objects. It describes the force per unit charge exerted by a charge or distribution of charges on a test charge placed within the field.
The electric field strength around a long, uniformly charged cylinder can be determined using Gauss's Law. Gauss's Law relates the electric field to the charge enclosed by a closed surface, simplifying calculations for symmetrical charge distributions.
For our cylinder with a uniform charge density of \( \lambda = 1.50 \times 10^{-9} \, \mathrm{C/m} \), the electric field outside the cylinder, at a radial distance \( r \), is given by:
\[E = \frac{\lambda}{2\pi \varepsilon_0 r} \]
This formula shows that the electric field decreases with increasing radial distance \( r \) from the cylinder's axis. The field's direction is radially outward, perpendicular to the cylinder's surface. When studying charged cylinders, it is important to note how the electric field behaves as it determines other related properties such as electric potential.

Equipotential surfaces are imaginary surfaces where every point on them represents the same electric potential. When it comes to a charged cylinder, these surfaces are particularly interesting.
For our long charged cylinder, the equipotential surfaces take the shape of cylindrical shells surrounding the charged cylinder. This is because the electric field lines radiate outward horizontally, making constant potential contours (shells) concentric around the cylinder.
If you imagine slicing the cylinder like bread, each slice would show a cross-section with rings representing different equipotential surfaces.
This configuration makes it easier to visualize and calculate changes in potential energy in the system. It also highlights that moving along an equipotential surface requires no work, as the electric potential remains constant at all points on that surface.

Electric potential, often simply called the potential, is a measure of the potential energy a unit charge would have at a specific point in an electric field. It allows us to determine the potential energy changes as charges move within an electric field.
For a point outside a charged cylinder, the electric potential \( V \) at a distance \( r \) can be found using the integral of the electric field:
\[V(r) = - \frac{\lambda}{2\pi \varepsilon_0} \ln \left( \frac{r}{R_0} \right)\]
Here, \( R_0 \) is the reference radial position, which is the surface of the cylinder. This formula reveals that the potential decreases logarithmically as you move away from the cylinder.
The negative sign indicates that as you move away from the reference point (cylinder surface), the potential decreases. Understanding electric potential helps predict how charges will behave when placed in the vicinity of charged objects like our cylinder.

Charge density is a measure of how much electric charge is accumulated over a given area or volume. In our scenario, we look at linear charge density, given by \( \lambda \). It tells us how much charge exists per meter length of the cylinder. In this exercise, charge density is \( 1.50 \, \mathrm{nC/m} \).
Having a uniform charge density means every segment of the cylinder has the same charge per unit length. This uniformity simplifies calculations, especially when using Gauss's Law to find the electric field or potential.
A key point to remember is that higher charge densities correspond to stronger electric fields and higher potentials near the surface of the charged object.
Charge density plays a central role in understanding the behavior of electric fields and potentials in systems involving charged objects. It gives us a foundation to predict how a system will behave under the influence of electric forces.