Problem 23 - University Physics with Modern Physics

Question

An electron is released from rest at a distance of 0.300 m from a large insulating sheet of charge that has uniform surface charge density +2.90 \(\times\) 10\(^{-12}\) C/m2. (a) How much work is done on the electron by the electric field of the sheet as the electron moves from its initial position to a point 0.050 m from the sheet? (b) What is the speed of the electron when it is 0.050 m from the sheet?

Step-by-Step Solution

Answer

(a) The work done is \(-6.56 \times 10^{-21}\) J. (b) The electron's speed is \(1.20 \times 10^5\) m/s.

Step 1: Understanding the Electric Field of an Infinite Sheet

The electric field (\(E\)) from an infinite sheet of charge with surface charge density (\(\sigma\)) is given by \[E = \frac{\sigma}{2\varepsilon_0}\] where \(\varepsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \ C^2/N \cdot m^2\)). Here, \(\sigma = +2.90 \times 10^{-12} \ C/m^2\).

Step 2: Calculating the Electric Field Strength

Plug the given values into the formula to find the electric field:\[E = \frac{2.90 \times 10^{-12}}{2 \times 8.85 \times 10^{-12}} = 0.164 \, N/C.\]

Step 3: Finding the Work Done by the Electric Field

The work done by the electric field on the electron as it moves from initial position to final position is \[W = qE \Delta x\] where \(q = -1.6 \times 10^{-19} \, C\) (charge of electron) and \(\Delta x = 0.250 \, m\) (the change in position, \(0.300 - 0.050\)). Thus,\[W = (-1.6 \times 10^{-19}) \cdot 0.164 \cdot 0.250 = -6.56 \times 10^{-21} \, J.\]

Step 4: Converting Work into Kinetic Energy

Since the work done by the electric field is converted into kinetic energy (\(\Delta KE\)) of the electron, we have \[\Delta KE = -W = 6.56 \times 10^{-21} \, J\].

Step 5: Calculating the Electron's Speed

The kinetic energy can be expressed as \(\Delta KE = \frac{1}{2}mv^2\) where \(m = 9.11 \times 10^{-31} \, kg\) is the mass of the electron. Solve for \(v\) to find the speed:\[v = \sqrt{\frac{2 \times 6.56 \times 10^{-21}}{9.11 \times 10^{-31}}} = 1.20 \times 10^5 \, m/s\].

Key Concepts

Work-Energy Principle

The work-energy principle is a fundamental concept in physics that relates the work done on an object to its change in kinetic energy. It essentially states that the work done by the sum of all forces acting on an object is equal to the change in that object's kinetic energy.

In the case of an electron moving in an electric field, any work performed by the electric field changes the electron's kinetic energy. So, when an electric field applies a force to an electron over a distance, this process transforms the work done on the electron into its kinetic energy.

• Work done by the electric field is calculated by the formula: \( W = qE \Delta x \), where \( q \) is the charge, \( E \) is the electric field strength, and \( \Delta x \) is the displacement in the direction of the field.
• This work results in a change in kinetic energy, thus \( \Delta KE = -W \), since the electron has negative charge.

In this scenario, the work-energy principle helps us understand how an initially stationary electron gains speed as it moves through an electric field. It's a beautiful example of energy transformation from electrical potential energy to kinetic energy, governed by the electric field.

Electron Motion

Understanding electron motion in an electric field is crucial in grasping the behavior of charged particles under electromagnetic forces. Electrons are negatively charged, and when placed in an electric field, they experience a force opposite to the direction of the field.

In this exercise, an electron is released from rest. Initially, since the electron isn't moving, its kinetic energy is zero. As it travels closer to the source of the field, the electric field performs work on the electron, increasing its kinetic energy.

• Since the electron has a negative charge, it moves in the opposite direction of the electric field lines.
• The force exerted by the electric field is given by \( F = qE \), causing the electron to accelerate.
• As the electron moves, its potential energy decreases, while its kinetic energy increases, keeping the total energy conserved.

Observing electron motion through an electric field provides insight into electron dynamics. This knowledge is essential in various applications, including electronics and particle physics, helping engineers and scientists to manipulate and use electrons effectively.

Kinetic Energy Calculations

Calculating kinetic energy involves understanding how energy changes as an object moves. In physics, kinetic energy is the energy an object possesses due to its motion, given by the formula \( KE = \frac{1}{2}mv^2 \). Here, \( m \) is the mass of the object, and \( v \) is its velocity.

For our electron, let's dive deeper into how we determine its speed. Once we know how much work the electric field has done, we can immediately translate this into kinetic energy.

• First, calculate work: \( W = -6.56 \times 10^{-21} \, J \) shows work done by the field.
• This work converts entirely into kinetic energy for the electron, \( \Delta KE = 6.56 \times 10^{-21} \, J \).
• Substitute into the equation \( \Delta KE = \frac{1}{2}mv^2 \) and solve for \( v \).

By simplifying the equation and solving, we find the speed of the electron. This calculated velocity gives us the precise measure of the electron's motion speed near the charged sheet. It's an elegant way to visualize how forces in an electric field result in tangible speeds and energies.