Motion of Charge Particles in Electric and Magnetic Field, Physics tutorial

Motion in an Electric Field:

Assume that a uniform electric field 'E' that is set up between two charged plates. Let us suppose a positive charge 'q' moving in the direction of the field having the velocity 'V'. You will remember that the force acting on the particle is represented by:

F = q E

The above represents that the force is independent of both the velocity and position of the particle. This constant force provides the particle a constant acceleration. From Newton's second law (F = mα), this constant acceleration is represented through

a = F/m = qE/m

Here, 'm' is the mass of the particle.

It obeys the above equation that the acceleration is in similar direction as the electric field. The equation as well illustrates that it is the ratio of charge to mass which finds out particle acceleration in a specific electric field. You can now understand why electrons that are much less in mass (around 2000 times) than protons however carrying similar charge, are readily accelerated in the electric fields. This is why most of the practical devices, such as television tubes, electron microscope and so on make use of the high accelerations that are possible with electrons.

Motion in a Magnetic Field:

Assume that a particle of mass 'm' moves in a circular orbit of radius 'ρ' having a constant speed 'v'. As we are familiar, that the acceleration of the particle is of magnitude mv2/ρ, and is for all time directed in the direction of the centre of the orbit. It follows that the acceleration is for all time perpendicular to the particle's instantaneous direction of motion.

We have observed that the force applied on a charged particle through a magnetic field is for all time perpendicular to its instantaneous direction of motion.

Assume that a particle of positive charge 'q' and mass 'm' moves in a plane perpendicular to a uniform magnetic field B. Assume that the particle moves, in an anti-clockwise manner, having constant speed 'v' (keep in mind that the magnetic field can't do work on the particle, therefore it can't influence its speed), in a circular orbit of radius 'ρ'. The magnetic force acting on the particle is of magnitude f = qvB and, according to the above equation this force is always directed in the direction of the centre of the orbit. Therefore, if

f = qvB = mv2

then we encompass a self-consistent picture. It obeys that:

ρ = mv/qB

The angular frequency of rotation of the particle (that is, the number of radians the particle rotates via in one second) is:

ω = v/ρ = qB/m

Cathode Ray Oscilloscope (CRO):

The force on a moving charge because of the magnetic field is employed to form pictures on a television screen. The major component of a television is the cathode-ray tube that is basically a vacuum tube in which the electric fields are employed to form a beam of electrons. This beam causes phosphor on the television screen to glow whenever struck by the electrons in the beam. With no magnetism, though, just the centre of the screen would be illuminated by the beam.

The CRO is mainly based on the given two principles:

1) If fast moving electrons hit the glass screen coated by zinc sulphide, they cause fluorescence.

2) As the mass of electrons is extremely small, they are simply deflected by the electric and magnetic fields and obey their variation by means of practically no time lag.

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An oscilloscope is a laboratory tool generally employed to display and analyzes the waveform of electronic signals. In consequence, the device sketches a graph of the instantaneous signal voltage as the function of time.

A typical oscilloscope can display alternating current (AC) or pulsating direct current (DC) waveforms having a frequency as low as approximately 1 hertz (Hz) or as high as several megahertz (MHz). High-end oscilloscopes can display signals having frequencies up to several hundred gigahertz (GHz). The display is broken up into so-called horizontal divisions (hor div) and vertical divisions (vert div). Time is displayed from left to right on the horizontal scale. Instantaneous voltage comes out on the vertical scale having positive values going upward and negative values going downward.

The oldest form of oscilloscope, still employed in a few labs today, is termed as the cathode-ray oscilloscope. It generates an image by causing a focused electron beam to travel, or sweep, in patterns across the face of a cathode ray tube (or CRT). More modern oscilloscopes electronically replicate the action of the CRT by employing a liquid crystal display (that is, liquid crystal display) identical to such found on notebook computers. The most complicated oscilloscopes use computers to process and display waveforms. Such computers can utilize any kind of display, comprising LCD, CRT and gas plasma.

Such days, typical high-end oscilloscopes are digital devices. They join to personal computers and utilize their displays. However these machines no longer make use of scanning electron beams to produce images of waveforms in the manner of the old cathode-ray scope, the fundamental principle is similar. Software regulates the sweep rate, vertical deflection and a host of other features that comprise:

  • Storage of waveforms for future reference and comparison
  • Display of some waveforms concurrently
  • Spectral analysis
  • Portability
  • Battery power option
  • Usability with all popular operating platforms
  • Zoom-in and zoom-out
  • Multi-color displays

Lorentz Force and its Applications:

If a particle having an electric charge like a proton, electron or ion (that is an atom having a surplus of electrical charges) moves, it forms a magnetic field around it. The similar is true for a wire having an electrical current flowing via it, as electrons are in motion via the wire.

Assume that a particle having a charge 'q' is moving having a velocity 'v' via a space, in which both the electric and magnetic fields exist concurrently, then the force applied on such a particle is represented by:

F = qE + q V ∧ B

The equation above is the vector sum of the electric force qE and the magnetic force q v ∧ B. It is termed as the Lorentz force equation and 'F' is the Lorentz force.

The Cyclotron:

The cyclotron was one of the first kinds of particle accelerators, and is still employed as the first phase of some big multi-phase particle accelerators. It makes utilization of the magnetic force on a moving charge to bend moving charges into the semicircular path between accelerations through an applied electric field. The applied electric field accelerates electrons among the 'dees' of the magnetic field area. The field is reversed at the cyclotron frequency to accelerate the electrons back across the gap.

How Cyclotrons Work:

Cyclotrons were a few of the first particle accelerators utilized to probe the inner workings of the elementary particles. The first cyclotron was invented in the year 1929 by Ernest Lawrence at the University of California at Berkeley.

Cyclotrons comprise of two hollow D-shaped electrodes (dees) sandwiched among a big dipole magnet. Charged particles, generally produced through a radioactive source at the center of the gap between the dees, are then drawn into one of the cavities through an electric field. A charge that moves to a magnetic field having direction perpendicular to the field will follow a semi-circular path. That is, the B field gives the centripetal force to curve the charged particle's orbit within the dees.

In order to accelerate the particles, the two D-shaped cavities should be driven at a constant frequency through a radio frequency (RF) accelerating power source that basically switches the charges on the dees backward and forward at just the right frequency to accelerate the particles across the gap. It is this switching of polarity at a particular frequency, termed as the cyclotron frequency which accelerates the beam of particles. As the beam spirals out, its frequency doesn't reduce, and it must continue to accelerate, as it is travelling more distance in similar time.

Finally, a charged plate at the outer edge of one of the dees deflects the path of the particles and switches them to a target at their maximum energy.

A moving charge in a cyclotron will move in the circular path beneath the affect of a constant magnetic field. When the time to complete one orbit is computed:

T = 2πr/v = 2πmv/qBv = 2πm/qB

Square wave of angular frequency ωCyclotron = qB/m

It is applied between the two sides of the magnetic poles; the charge will be boosted again at just right time to accelerate it across the gap. Therefore the constant cyclotron frequency can carry on accelerating the charge (so long as it is not relativistic).

Uses for Cyclotrons:

For some decades, cyclotrons were the best source of high-energy beams for the nuclear physics experiments; some of the cyclotrons are still in use for this kind of research. Cyclotrons can be employed to treat cancer. Ion beams from cyclotrons can be employed, as in proton therapy, to go through the body and kill tumors by radiation damage, whereas minimizing damage to healthy tissue all along their path. Cyclotron beams can be employed to bombard other atoms to generate short-lived positron-emitting isotopes appropriate for PET imaging.

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