91Ó°ÊÓ

An electric field represents a region in space where electric charges experience a force. For a line of charge, the electric field behaves differently than for a point charge. Here, the electric field due to a very long line of charge with uniform linear charge density \lambda\ is given by the formula: \( E = \frac{\lambda}{2\pi\epsilon_0 r} \). This illustrates how the field strength decreases with increasing distance \( r \) from the line of charge.

\[ E \] denotes the electric field strength, \( \lambda \) is the linear charge density (charge per unit length), and \( \epsilon_0 \) is the vacuum permittivity. The electric field is crucial in determining how a charged particle, like the sphere in our problem, will move under the influence of forces.

A line of charge refers to a configuration where charge is distributed uniformly along an infinitely long line. This differs from point charges, leading to unique electric field properties.

When dealing with a line of charge, the electric field at a point depend only on the perpendicular distance from the line rather than the distance from a single charge point. The formula \( E = \frac{\lambda}{2\pi\epsilon_0 r} \) highlights this relationship.

Such forms of charge distribution often simplify calculations and are key in understanding how electric fields can be applied in different scenarios. A practical example includes cables or charged rods that we encounter in real-world applications.

Coulomb's Law describes the force between two charges. Essentially, it states that the force between static charges is proportional to the product of their charges and inversely proportional to the square of the distance between them.

While Coulomb’s Law is directly used for point charges, its principles help understand forces from lines of charge. The electric field termed earlier builds upon these basics to cater to line charges.

Understanding this law is essential as it forms the groundwork for evaluating electrical forces in many systems, allowing us to predict how charged particles will move under given conditions.

Electric potential energy is the energy a charge possesses due to its position in an electric field. In the case of a charged sphere moving in a field generated by a line of charge, potential energy changes as it moves closer or farther away.

The potential energy difference is calculated using \( \Delta U = q \cdot (V_{final} - V_{initial}) \) where \( q \) is the charge of the sphere, and \( V \) the potential at given points. Here, \( V \) in terms of a line of charge is given by \( V = -\frac{\lambda}{2\pi\epsilon_0}\ln\left(\frac{r}{r_0}\right) \).

This energy transformation is pivotal: when the sphere starts from rest, its initial kinetic energy is zero, so the entire change in potential energy converts to kinetic energy. Thus, knowing this difference gives us the kinetic energy of the sphere at its new position.