An electron is acted upon by a force of 5.00 ×10^-15 N due to an electric field. Find the accele

step 1 This is chapter 37 problem number 30. We're given an electron. And due to an electric field, the

An electron is acted upon by a force of 5.00 ×10^-15 N due...

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

Key Concepts

Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the rate of change of its momentum. In the simplest case of constant mass, this relationship is expressed as F = ma, where a is the acceleration produced by force F acting on mass m. This concept underpins how forces produce changes in motion in classical mechanics.

Electron Mass

The electron mass is a fundamental property of electrons, used in calculations involving their motion under applied forces. In problems involving the acceleration of electrons, the rest mass is used to determine the resulting acceleration when a known force is applied, making it a critical parameter in both classical and relativistic dynamics.

Relativistic Dynamics

Relativistic dynamics extends classical mechanics to situations where objects move at speeds close to the speed of light. At these high speeds, the effective inertia of an object increases due to relativistic effects, and the simple form of F = ma must be modified to incorporate the effects of the Lorentz factor. This concept is key to accurately describe the behavior of particles like electrons moving at relativistic speeds.

Lorentz Factor

The Lorentz factor, denoted by ?, is a factor that accounts for relativistic effects such as time dilation and length contraction when objects move at speeds near the speed of light. It is defined as ? = 1/?(1 - (v/c)²), where v is the object's speed and c is the speed of light. In dynamics, the Lorentz factor adjusts the relationship between force and acceleration, especially when the force is applied in the direction of motion.

Transcript

00:01 This is chapter 37 problem number 30.

00:02 We're given an electron.

00:05 And due to an electric field, there is the force acting on this electron and the magnitude of 5 times center per 0 .50 moutons.

00:14 And we know the mass of the electron 9 .1 .1 times 10 per negative 31 kilograms.

00:20 So we're asked to calculate the acceleration on this electron if a, speed of this electron is one kilometer per second, which is a thousand meters per second, which is not a relativistic speed.

00:46 So you new 20 mechanics is going to be sufficient enough and times a equals the force.

00:54 If you're after the acceleration, then if we divide both sides by the mass, ms are going to cancel.

01:01 The acceleration is going to be equal to force.

01:03 Divided by the mass forces 5 times 10 to per of negative 15 newtons.

01:09 The mass is 9 .11 times 10 to per of negative 31 kilograms.

01:14 In this case, our acceleration is going to be 5 .49 times 10 to per 15 meters per second squared.

01:22 It's a very high acceleration.

01:25 Part b, if again, find acceleration if now we're dealing with the relativistic speeds, 2 .5.

01:33 Times 10 to curve 8 meters per second.

01:37 And there is an important part to this problem where it says f is parallel with respect to the velocity vector because this is going to change everything...

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