Chapter 29: Q.67 (page 834)

If A particle of charge $q$ and mass $m$ moves in the uniform fields $\vec{E} = E_{0} \hat{k} and \vec{B} = B_{0} \hat{k}$ . At $t = 0$ , the particle has velocity ${\vec{v}}_{0} = v_{0} \hat{i}$ . What is the particle's speed at a later time $r$ ?

Short Answer

Expert verified

The particle's speed at a later time $v = \sqrt{v_{o}^{2} + {(\frac{q E_{o} t}{m})}^{2}}$

Step by step solution

Given Values

$\vec{E} = E_{o} \hat{k}$

$\vec{B} = B_{o} \hat{k}$

$\vec{v} = v_{o} \hat{i}$

$t = 0$

We use the equations,

${\vec{F}}_{E} = q \vec{E} = q E_{o} \hat{k}$

$\vec{F_{M}} = \vec{q v} \times \vec{B} = q v_{o} B_{o} (- j)$

In the x-y plane, for circular movement:

Equation Solving

$\vec{F_{M}} = q v_{o} B_{o} \hat{r} ⟺ F_{E} = q E_{o} \hat{k}$

$\sum \vec{F} = m \vec{a}$

$F_{1 z} = q E_{o} - m a_{z}$

$F_{1 r} = q v_{o} B_{o} = m a_{τ} = \frac{m v_{o}^{2}}{2}$

$F_{1 t} = 0$

Final Equation

$we find \vec{v} (t) :$

$\vec{v} (t) = v_{o} \hat{r} + (0 + a_{z} t) \hat{k}$

$\vec{v} (t) = v_{o} \hat{i} + a_{z} t \hat{k}$

$v = \sqrt{v_{o}^{2} + {(a_{z} t)}^{2}}$ $(a_{z} = \frac{q E_{o}}{m} \cdot)$

$v = \sqrt{v_{o}^{2} + {(\frac{q E_{o} t}{m})}^{2}}$

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If A particle of charge $q$ and mass $m$ moves in the uniform fields $\vec{E} = E_{0} \hat{k} and \vec{B} = B_{0} \hat{k}$ . At $t = 0$ , the particle has velocity ${\vec{v}}_{0} = v_{0} \hat{i}$ . What is the particle's speed at a later time $r$ ?

Short Answer

Step by step solution

Given Values

Equation Solving

Final Equation

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If A particle of charge qand mass mmoves in the uniform fields E→=E0k^andB→=B0k^. At t=0, the particle has velocity v→0=v0i^. What is the particle's speed at a later time r ?

Short Answer

Step by step solution

Given Values

Equation Solving

Final Equation

One App. One Place for Learning.

Most popular questions from this chapter

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If A particle of charge $q$ and mass $m$ moves in the uniform fields $\vec{E} = E_{0} \hat{k} and \vec{B} = B_{0} \hat{k}$ . At $t = 0$ , the particle has velocity ${\vec{v}}_{0} = v_{0} \hat{i}$ . What is the particle's speed at a later time $r$ ?