Problem 26 - College Physics

Question

An electron is acted upon by a force of \(5.00 \times 10^{-15} \mathrm{N}\) due to an electric field. Find the acceleration this force produces in each case: (a) The electron's speed is 1.00 \(\mathrm{km} / \mathrm{s}\) . (b) The electron's speed is \(2.50 \times 10^{8} \mathrm{m} / \mathrm{s}\) and the force is parallel to the velocity.

Step-by-Step Solution

Answer

Acceleration is \(5.49 \times 10^{15} \text{ m/s}^2 \) in both cases.

Step 1: Identify Known Values

The force acting on the electron is given as \(F = 5.00 \times 10^{-15} \text{ N}\). The mass of an electron is \(m = 9.11 \times 10^{-31} \text{ kg}\). Two cases are given: (a) Speed \(v = 1.00 \text{ km/s} = 1000 \text{ m/s}\), (b) Speed \(v = 2.50 \times 10^{8} \text{ m/s}\).

Step 2: Use Newton's Second Law for Case (a)

Newton's second law is \(F = ma\). For case (a), rearrange to find the acceleration \(a\): \(a = \frac{F}{m}\). Substitute the values: \(a = \frac{5.00 \times 10^{-15}}{9.11 \times 10^{-31}}\).

Step 3: Calculate Acceleration for Case (a)

Calculating the acceleration gives: \(a = 5.49 \times 10^{15} \text{ m/s}^2 \).

Step 4: Use Newton's Second Law for Case (b)

For an electron moving at high speed, Newton's second law in its classical form \(F = ma\) still applies as the speed is less than the speed of light. Use \(a = \frac{F}{m}\). Substitute the given values: \(a = \frac{5.00 \times 10^{-15}}{9.11 \times 10^{-31}}\).

Step 5: Calculate Acceleration for Case (b)

The result of the calculation is the same as in case (a): \(a = 5.49 \times 10^{15} \text{ m/s}^2 \), assuming classical mechanics is applicable without relativistic effects.

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that relates the motion of an object to the forces acting on it. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The formula is given as:

• \( F = ma \)

Here, \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is the acceleration. This equation helps us determine how an object will move when subjected to different forces.
To find the acceleration, rearrange the formula to \( a = \frac{F}{m} \). This calculation illustrates that heavier objects require more force to accelerate at the same rate as lighter ones.
Understanding this principle is crucial for solving problems related to motion, such as determining the effects of forces on moving objects.

Electron Mass

The electron is a subatomic particle with a very small mass. Its mass is approximately \( 9.11 \times 10^{-31} \text{ kg} \).
Because of its tiny mass, an electron can experience significant accelerations when even small forces are applied.
This characteristic is important in many physical phenomena and technological applications, such as in cathode ray tubes or beam steering in electron microscopes.
In problems involving electrons, the mass of the electron plays a critical role in calculating the effects of forces. When you apply a force to an electron, you must consider its small mass to accurately compute its acceleration using Newton's Second Law.
This highlights why electrons can attain high accelerations even with relatively small forces, making them essential particles in fields like electronics and quantum mechanics.

Electric Force

Electric force is the interaction between charged particles due to their electric fields. It can cause objects to attract or repel each other, depending on their charges.
In this exercise, an electric force of \( 5.00 \times 10^{-15} \text{ N} \) acts on an electron. This force results from external electric fields interacting with the electron's charge, causing it to move or accelerate.
Electric forces are fundamental in various scientific and engineering disciplines, from telecommunications to semiconductors. In physics exercises, understanding how these forces interact with particles like electrons is crucial for solving problems related to electric fields, currents, and circuits.

• Electric force is governed by Coulomb's law and can be calculated using equations based on the charges involved and the distance between them.

This understanding enables us to predict how charged particles behave under different electric forces and fields.

Velocity Impact on Acceleration

Velocity refers to the speed and direction of a moving object. While velocity itself doesn't directly affect acceleration, velocities can impact how forces, like electric forces, produce acceleration.
In the original exercise, the electron's velocity was considered in two cases, but the acceleration was determined by the force and the electron's mass.

• When dealing with high velocities, close to the speed of light, special relativity must be considered, affecting mass and therefore acceleration. However, in this exercise, such relativistic effects are negligible.
• It's important to note that the speed of the electron in these cases does not alter the calculated acceleration due to the constant force and mass.

This exercise exemplifies how, in a classical physics approach, velocity primarily informs movement dynamics rather than directly influencing the force-to-acceleration relationship in Newton's Second Law.