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6) Consider a magnetic field given as $\vec{B} = B_0 \hat{k}$ where $B_0$ is a constant. A particle of charge $q_0$ and inertial mass $m_0$ starts its motion $\vec{r}(0) = 0$ and $\vec{v}(0) = v_{0}\hat{j}$. Using the Lorentz force given as $\vec{F} = q_0 \vec{v} \times \vec{B}$, you're to study the motion. That is find a) $\vec{v}(t)$ and $\vec{r}(t)$ and $\vec{a}(t)$ and show that the particle performs uniform circular motion, with radius $R_0$ and angular frequency $\omega_0$. b) Find a relation between $R_0$ and $\omega_0$. What do you see? c) Discuss the limit $v_0 << c$ and compare it to the ultrarelativistic case you've. Write a short paragraph about the results you found and what you individually expected from the outset. (when you first read the problem).

          6) Consider a magnetic field given as $\vec{B} = B_0 \hat{k}$ where $B_0$ is a constant. A particle of charge $q_0$ and inertial mass $m_0$ starts its motion $\vec{r}(0) = 0$ and $\vec{v}(0) = v_{0}\hat{j}$. Using the Lorentz force given as $\vec{F} = q_0 \vec{v} \times \vec{B}$, you're to study the motion.

That is find
a) $\vec{v}(t)$ and $\vec{r}(t)$ and $\vec{a}(t)$ and show that the particle performs uniform circular motion, with radius $R_0$ and angular frequency $\omega_0$.
b) Find a relation between $R_0$ and $\omega_0$. What do you see?
c) Discuss the limit $v_0 << c$ and compare it to the ultrarelativistic case you've.

Write a short paragraph about the results you found and what you individually expected from the outset. (when you first read the problem).

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6 consider a magnetic field given as vecb b0 hatk where b0 is a constant a particle of charge q0 and inertial mass m0 starts its motion vecr0 0 and vecv0 v0hatj using the lorentz force 00016

Added by Juan Carlos S.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sam Stansfield

03/10/2024

Step 1

Since $\vec{B} = B_0 \hat{k}$, we have $\vec{F} = q_0 \vec{v} \times B_0 \hat{k} = q_0 B_0 (\vec{v} \times \hat{k})$ The equation of motion is given by $\vec{F} = m_0 \vec{a}$, where $\vec{a} = \frac{d\vec{v}}{dt}$. Therefore, $m_0 \frac{d\vec{v}}{dt} = q_0 B_0 Show more…

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6) Consider a magnetic field given as $\vec{B} = B_0 \hat{k}$ where $B_0$ is a constant. A particle of charge $q_0$ and inertial mass $m_0$ starts its motion $\vec{r}(0) = 0$ and $\vec{v}(0) = v_{0}\hat{j}$. Using the Lorentz force given as $\vec{F} = q_0 \vec{v} \times \vec{B}$, you're to study the motion. That is find a) $\vec{v}(t)$ and $\vec{r}(t)$ and $\vec{a}(t)$ and show that the particle performs uniform circular motion, with radius $R_0$ and angular frequency $\omega_0$. b) Find a relation between $R_0$ and $\omega_0$. What do you see? c) Discuss the limit $v_0 << c$ and compare it to the ultrarelativistic case you've. Write a short paragraph about the results you found and what you individually expected from the outset. (when you first read the problem).

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Transcript

00:01 Okay, so we're given the equation of motion for a charged particle in an electromagnetic field.

00:09 We're told what the components of electric and magnetic field look like.

00:13 They're constant.

00:14 We're also given our initial velocity, which is in the z direction.

00:20 So the first thing we want to do is we want to calculate.

00:25 We want to write down the equations.

00:27 Equations.

00:28 Okay, so we can figure out this cross product by taking the determinant.

00:35 Okay, so we get this equation, and let's look at the vector components of it.

01:27 So this equation here, the y equation, tells us that y dot is a constant.

01:35 That's our initial velocity in the y direction, which is zero because our initial velocity is in the z direction.

01:48 So that means we've got zero velocity in the y direction all along.

01:58 So that means that the motion is in the xz plane.

02:10 Okay, so let's take this equation, and i can integrate it because we just got time derivatives here.

02:18 So we get m times z dot equals qbx plus some constant, let's call it c1.

02:32 So we'll get more constants as we integrate these things.

02:37 Okay, so that tells me what z dot is, and i can substitute that back into this equation.

03:04 And then it's z dot, which is, okay, okay, now i'll bring the x terms over to the left.

03:36 And first of all, i'll divide by m.

03:38 So i get whatever c1 might be.

04:12 So the way this goes is we solve this equation for x.

04:19 And then once we got the answer, we plug it back up in here.

04:23 And we integrate this equation and find z.

04:28 So let's look now at part a.

04:34 So now we're going to say what if v zero is e over b.

04:43 So if v zero is e over b, then it's easy to see that our force, that's equal to zero...

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