# Particle moving perpendicular to a constant magnetic field

A particle of mass m and charge q is moving with speed v perpendicular to a constant magnetic field B(vector)=BoZ-hat

a) What is the acceleration of the particle? Calculate the x, y and z-component of a as a function of time. You may choose the direction of the x and y-axes for your own convenience. Describe the particle's path as it moves under the influence of the magnetic field.

b) Find the power radiated per unit solid angle by the particle, assuming v << c.

c) Time average your result in part (b). (Just set up the expression for time average).

https://brainmass.com/physics/lorentz-force-law/particle-moving-perpendicular-to-a-constant-magnetic-field-109871

#### Solution Preview

Please see the attached file.

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Here is the plain TEX source

magnification=magstep1

baselineskip=12pt

parindent = 0pt

parskip = 12pt

defl{left}

defr{right}

defla{langle}

defra{rangle}

defp{partial}

centerline{bf Charged particle moving in a uniform magnetic field.}

As stated in part (b) we assume non-relativistic motion with $vll c$.

The ``Introduction to Electrodynamics" by David Griffiths will be referred to as

bf ``the book"rm.

bigskip

bf (a) rm

A particle with charge $q$ moving in a uniform magnetic field $vec B = B_ohat z$

is accelerated by the Lorentz force

$$

vec F = qvec v times vec B,

eqno(a.1)

$$

see equation (5.1) of the book.

Therefore its equation of motion (second law of Newton),

$$

vec a = vec F/m

eqno(a.2)

$$

where $m$ is its mass and $vec a$ is its acceleration,

can be written in Cartesian $(x,y,z)$ components as

$$

eqalign{

dot v_x &= {qB_oover m} v_y,cr

dot v_y &= -{qB_oover m} v_x,cr

dot v_z &= 0.cr

}

eqno(a.3)

$$

From the last of equations (a.3) we see that $v_z = const$ and the first two of

equations (a.3) can also be written as

$$

dot {vec v_bot} = -omega_o hat z times vec v_bot ,

eqno(a.4)

$$

where the Larmor frequency is

$$

omega_o = {qB_oover m}

eqno(a.5)

$$

and $vec v_bot = (v_x, v_y,0)$.

Equation (a.4) or equivalently the first two of equations (a.3)

describe a circular motion in ``$vec v$-space" with frequency $omega_o$ ...

#### Solution Summary

It investigates the motion of the charge in a constant magnetic field.

Paths Through a Uniform Magnetic Field, Solenoids and Net Magnetic Field

1) The figure (*see attachment) shows 11 paths through a region of uniform magnetic field. One path is a straight line; the rest are half circles. The table (*see attachment) gives the masses, charges and speeds of the 11 particles that take these paths through the field in the direction shown. Which path corresponds to which particle in the table?

2) The following table (*see attachment) gives the number of turns per unit length, n, and the current, I, through six ideal solenoids of different radii. You want to combine them concentrically to produce a net magnetic field of zero along the central axis. Can this be done with (a) two of them, (b) three of them, (c) four of them, (d) five of them, (e) six of them? If so, answer by listing which solenoids are to be used and indicate the direction of the currents.

3) Two long parallel wires which are separated by a distance of 18 cm carry currents of 7A and 8A. (a) Where is the net magnetic field zero? (i.e. how many cm from the midpoint between the two wires is the magnetic field=zero?) (b) what is the B-field at a point 5cm to the left of wire 1 and 5cm below both wires?

4) In the figure (*see attached), E = 100 V, R1 = 20 ohms, R3 = 30 ohms and L = 2 H. Find the values of I1 and I2 (a) immediately after closing the switch S, (b) a long time after switch S is closed, (c) immediately after switch S is opened and (d) a long time after switch S is opened.

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