Problem 35 - University Physics with Modern Physics

Question

A very small sphere with positive charge \(q=+8.00 \mu \mathrm{C}\) is released from rest at a point 1.50 \(\mathrm{cm}\) from a very long line of uniform linear charge density \(\lambda=+3.00 \mu \mathrm{Cm} .\) What is the kinetic energy of the sphere when it is 4.50 \(\mathrm{cm}\) from the line of charge if the only force on it is the force exerted by the line of charge?

Step-by-Step Solution

Answer

The kinetic energy when the sphere is 4.50 cm from the line is due to the change in electric potential energy, which is converted into kinetic energy.

Step 1: Understand the Problem

We want to find the kinetic energy of a charged sphere after it has moved a certain distance in an electric field created by a line of charge. We start by noting the initial and final distances of the sphere from the line and that the sphere begins at rest.

Step 2: Identify Key Formulas

The force on a charge in an electric field, created by a line of charge, is given by Coulomb's Law. However, since this is a line of charge, we use the electric field from a line, given by: \( E = \frac{\lambda}{2\pi\epsilon_0 r} \). The change in electric potential energy though, which relates to kinetic energy, uses the potential difference formula.

Step 3: Calculate Initial and Final Electric Potential

The electric potential at a distance \( r \) from an infinite line of charge is \( V = -\frac{\lambda}{2\pi\epsilon_0}\ln\left(\frac{r}{r_0}\right) \). Calculate the potential at the initial point (1.5 cm) and the final point (4.5 cm).

Step 4: Compute Change in Electric Potential Energy

Using the initial and final potentials, calculate the change in electric potential energy, \( \Delta U = q \cdot (V_{final} - V_{initial}) \).

Step 5: Convert Potential Energy to Kinetic Energy

Since the sphere starts from rest, its initial kinetic energy is zero. Thus, the change in electric potential energy is equal to the kinetic energy gained, \( \Delta U = K \).

Step 6: Solve the Calculation

Compute the values using \( \lambda = 3.00 \underline{\phantom{xxx}}\mu\text{C/m}, \; q = 8.00 \underline{\phantom{xxx}}\mu\text{C}, \; \epsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/(\text{N} \cdot \text{m}^2) \). Plug these values into the formulas, to find the final kinetic energy.

Key Concepts

Electric Field

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.

Line of Charge

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

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

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.