A particle is said to be nonrelativistic as long as its...

A particle is said to be nonrelativistic as long as its speed is less than one-tenth the speed of light, or less than $3.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$. (a) How long will an electron remain nonrelativistic if it starts from rest in a region of an electric field of $1.00 \mathrm{~N} / \mathrm{C} ?(\mathrm{~b})$ How long will a proton remain nonrelativistic in the same electric field? (c) Electric fields are commonly much larger than $1 \mathrm{~N} / \mathrm{C}$. Will the charged particle remain nonrelativistic for a shorter or a longer time in a much larger electric field?

A particle is said to be nonrelativistic as long as its speed is less than one-tenth the speed of light, or less than $3.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$. (a) How long will an electron remain nonrelativistic if it starts from rest in a region of an electric field of $1.00 \mathrm{~N} / \mathrm{C} ?(\mathrm{~b})$ How long will a proton remain nonrelativistic in the same electric field?
(c) Electric fields are commonly much larger than $1 \mathrm{~N} / \mathrm{C}$. Will the charged particle remain nonrelativistic for a shorter or a longer time in a much larger electric field?

Key Concepts

Nonrelativistic Approximation

This concept involves treating the motion of a particle with speeds much less than the speed of light using classical mechanics. The approximation implies that relativistic corrections (those based on Einstein’s theory of relativity) are negligible, simplifying calculations such as acceleration and velocity over time.

Electric Field and Force on a Charged Particle

The electric field exerts a force on a charged particle equal to the charge multiplied by the field strength. This force produces an acceleration on the particle according to Newton’s second law. Understanding this relationship is essential for analyzing the particle’s motion in a uniform electric field.

Kinematics Under Constant Acceleration

Under the influence of a constant acceleration, kinematic equations describe how a particle’s speed changes over time. In particular, one can calculate the time required for the particle to reach a certain speed starting from rest by using the relation between acceleration, time, and velocity.

Relativistic Speed Threshold

A relativistic threshold is set when a particle’s speed becomes a significant fraction of the speed of light. In problems like these, a speed less than one-tenth the speed of light is defined as nonrelativistic. This threshold helps determine the regime in which classical equations remain valid.

Impact of Particle Mass

The mass of a particle directly affects its acceleration under a given force. Lighter particles (like electrons) accelerate much more quickly compared to heavier ones (like protons) when subjected to the same electric field. This difference in acceleration leads to different timescales for reaching the relativistic regime.

Transcript

00:01 In this problem, we're going to explore how long it takes for an electron and a proton to reach a relativistic speed, which we consider to begin at three times 10 to the seventh meters per second or one -tenth the speed of right.

00:14 So let's start looking at the electron.

00:17 I want to keep everything positive, so i'm going to choose my electric field point in the negative x direction.

00:25 That gives me the direction of the force on a positive charge, but electron is negatively charged, so do experience a force.

00:33 In the positive x direction to the right.

00:36 Now, with that, assuming that's the only force acting on the object, our f -net, which is ma, and let's not forget the general formula for the force on a charge in electric field is given by that charge times the electric field.

00:57 That's how we find the force.

00:59 So in this case, we got a charge of minus e times the electric field.

01:06 And from that we can get the acceleration, ax, is going to be minus e times the x component of electric field, which is minus capital e, because it's in the negative x direction, over the mass of the electron.

01:19 So that's the magnitude of the charge, magnitude of the field over m.

01:27 So we have our acceleration.

01:29 Now what form do we want to use to find the time? well, we can use this formula.

01:33 And i'll write v -relativistic x, that's the 3 times 10 of the seventh.

01:39 This object starts, the electron starts from rest plus a x.

01:44 I'm going to start this object at t equals zero, so i'm just going to look for what i call t relativistic, the time when it actually now reaches the relativistic speed.

01:55 V -0x is zero, so this term is gone.

01:59 So that tells me immediately, the time it takes to get to the relativistic speed is going to be be relativistic x over a x and we have a formula for a x e e over the mass electron and now we can put in our numbers three times 10 to the 7 meters per second the mass the electron that comes to the top 9 .11 times 10 to minus 31 kilograms.

02:43 Magnet to the electric charge is 1 .6 times 10 to the minus 19.

02:50 Coolomes.

02:51 The electric field they give us 1 newton per cooler.

02:55 And this works out to be 1 .71 times 10 to the minus 4 seconds.

03:03 So pretty reaching of that relativeistic speed, what you expect? the electrons got very small mass.

03:15 So that was part a.

03:18 Part b is for the proton.

03:25 And i'll draw a picture again just to reinforce certain ideas.

03:31 I'll have my electric field pointing to the right this time.

03:35 Here's my proton...