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Motion of a Charged Particle in a Uniform Magnetic Field

Uniform electric and magnetic fields with strength E and B are

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Medium

Solving time: 3 mins

Uniform electric and magnetic fields with strength $E$ and $B$ are directed along the $y$ -axis. A particle with specific charge $q / m$ leaves the origin in the direction of $x$ -axis with an initial velocity $v_{0}$ . Find
(a) the $y$ -coordinate of the particle when it crosses the $y$ -axis for nth time.
(b) the angle $α$ between the particle's velocity vector and the $y$ -axis at that moment.

Views: 6,016 students

Book:

Electricity and Magnetism (DC Pandey)

Text solutionVerified

As discussed in Type-9 path of the particle is a helix of increasing pitch. The axis of the helix is parallel to

y

-axis (parallel to

E

) and plane of circle of the helix is

x z

(perpendicular to

B

). The particle will cross the

y

-axis after time,

t = n T = n (\frac{2 πm}{Bq}) = \frac{2 πmn}{Bq}

The

y

-coordinate of particle at this instant is

y = \frac{1}{2} a_{y} t^{2}

where,

a_{y} = \frac{F _{y}}{m} = \frac{qE}{m}

y = \frac{1}{2} (\frac{qE}{m}) (\frac{2 πmn}{Bq})^{2}

= \frac{2 n ^{2} m E π ^{2}}{q B ^{2}}

(b) At this moment

y

-component of its velocity is

[v_{y} = a_{y} t = (\frac{qE}{m}) (\frac{2 πmn}{Bq}) = 2 πn (\frac{E}{B})]

moment is

α = tan^{- 1} (\frac{v _{x z}}{v _{y}})

Here,

v_{x z} = v_{x}^{2} + v_{z}^{2} = v_{0}

α = tan^{- 1} (\frac{B v _{0}}{2 πn E})

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Uniform electric and magnetic fields with strength

E

and

B

are directed along the

y

-axis. A particle with specific charge

q / m

leaves the origin in the direction of

x

-axis with an initial velocity

v_{0}

. Find
(a) the

y

-coordinate of the particle when it crosses the

y

-axis for nth time.
(b) the angle

α

between the particle's velocity vector and the

y

-axis at that moment.

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Question Text	Uniform electric and magnetic fields with strength $E$ and $B$ are directed along the $y$ -axis. A particle with specific charge $q / m$ leaves the origin in the direction of $x$ -axis with an initial velocity $v_{0}$ . Find (a) the $y$ -coordinate of the particle when it crosses the $y$ -axis for nth time. (b) the angle $α$ between the particle's velocity vector and the $y$ -axis at that moment.
Topic	Moving Charges and Magnetism
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	68

Text solutionVerified

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