in a cyclotron, as the proton spirals outward, what happens to the frequency of circling?

Learning Objectives

By the end of this department, you will be able to:

• Describe the effects of a magnetic field on a moving charge.
• Calculate the radius of curvature of the path of a charge that is moving in a magnetic field.

Magnetic force can cause a charged particle to move in a circular or spiral path. Cosmic rays are energetic charged particles in outer space, some of which arroyo the Earth. They can be forced into spiral paths past the Earth's magnetic field. Protons in behemothic accelerators are kept in a circular path past magnetic strength. The chimera chamber photograph in Figure ane shows charged particles moving in such curved paths. The curved paths of charged particles in magnetic fields are the basis of a number of phenomena and can fifty-fifty be used analytically, such equally in a mass spectrometer.

Effigy 1. Trails of bubbles are produced by loftier-energy charged particles moving through the superheated liquid hydrogen in this artist'south rendition of a chimera sleeping room. At that place is a strong magnetic field perpendicular to the page that causes the curved paths of the particles. The radius of the path can be used to detect the mass, accuse, and free energy of the particle.

So does the magnetic force cause circular motion? Magnetic force is always perpendicular to velocity, then that it does no work on the charged particle. The particle's kinetic energy and speed thus remain abiding. The direction of movement is affected, but non the speed. This is typical of compatible round motion. The simplest case occurs when a charged particle moves perpendicular to a uniform B-field, such as shown in Figure 2. (If this takes identify in a vacuum, the magnetic field is the ascendant factor determining the movement.) Hither, the magnetic forcefulness supplies the centripetal force F _c= mv ²/r. Noting thatsin θ = 1 , we encounter that F = qvB .

Figure 2. A negatively charged particle moves in the plane of the page in a region where the magnetic field is perpendicular into the page (represented past the small circles with x'southward—similar the tails of arrows). The magnetic strength is perpendicular to the velocity, and and then velocity changes in direction just non magnitude. Uniform round motion results.

Considering the magnetic force F supplies the centripetal force F _c, we have

[latex]qvB=\frac{mv^{2}}{r}\\[/latex].

Solving for r yields

[latex]r=\frac{mv}{qB}\\[/latex].

Here, r is the radius of curvature of the path of a charged particle with mass grand and charge q, moving at a speed v perpendicular to a magnetic field of strength B. If the velocity is non perpendicular to the magnetic field, and so v is the component of the velocity perpendicular to the field. The component of the velocity parallel to the field is unaffected, since the magnetic forcefulness is zilch for move parallel to the field. This produces a screw movement rather than a circular ane.

Example i. Computing the Curvature of the Path of an Electron Moving in a Magnetic Field: A Magnet on a Boob tube Screen

A magnet brought near an old-fashioned Tv set screen such as in Figure 1 (Boob tube sets with cathode ray tubes instead of LCD screens) severely distorts its flick by altering the path of the electrons that brand its phosphors glow. (Don't try this at habitation, as it will permanently magnetize and ruin the TV.) To illustrate this, calculate the radius of curvature of the path of an electron having a velocity of 6 . 00 × ten ⁷ thousand/s  (corresponding to the accelerating voltage of about x.0 kV used in some TVs) perpendicular to a magnetic field of strength B= 0.500 T (obtainable with permanent magnets).

Figure 1. Side view showing what happens when a magnet comes in contact with a computer monitor or TV screen. Electrons moving toward the screen spiral about magnetic field lines, maintaining the component of their velocity parallel to the field lines. This distorts the image on the screen.

Strategy

Nosotros tin find the radius of curvature r directly from the equation [latex]r=\frac{mv}{qB}\\[/latex], since all other quantities in it are given or known.

Solution

Using known values for the mass and charge of an electron, along with the given values of v and B gives united states of america

[latex]\begin{array}{lll}r=\frac{mv}{qB}& =& \frac{\left(ix.xi\times{10}^{-31}\text{ kg}\right)\left(six.00\times 10^{vii}\text{ thousand/southward}\correct)}{\left(one.60\times\text{ten}^{-19}\text{ C}\right)\left(0.500\text{ T}\correct)}\\ & =& 6.83\times {10}^{-4}\text{ one thousand}\end{assortment}\\[/latex]

or

r= 0.683 mm.

Discussion

The small radius indicates a large event. The electrons in the TV motion picture tube are made to move in very tight circles, greatly altering their paths and distorting the image.

Figure 2 shows how electrons not moving perpendicular to magnetic field lines follow the field lines. The component of velocity parallel to the lines is unaffected, and so the charges screw along the field lines. If field forcefulness increases in the direction of move, the field will exert a force to boring the charges, forming a kind of magnetic mirror, as shown beneath.

Figure 2. When a charged particle moves forth a magnetic field line into a region where the field becomes stronger, the particle experiences a force that reduces the component of velocity parallel to the field. This strength slows the motion forth the field line and here reverses information technology, forming a "magnetic mirror."

The backdrop of charged particles in magnetic fields are related to such different things as the Aurora Australis or Aurora Borealis and particle accelerators. Charged particles approaching magnetic field lines may become trapped in spiral orbits virtually the lines rather than crossing them, as seen to a higher place. Some cosmic rays, for instance, follow the Earth's magnetic field lines, entering the atmosphere near the magnetic poles and causing the southern or northern lights through their ionization of molecules in the atmosphere. This glow of energized atoms and molecules is seen in Effigy 1 on page. Those particles that arroyo middle latitudes must cross magnetic field lines, and many are prevented from penetrating the atmosphere. Cosmic rays are a component of background radiation; consequently, they give a higher radiation dose at the poles than at the equator.

Figure 3. Energetic electrons and protons, components of cosmic rays, from the Sun and deep outer infinite often follow the Globe's magnetic field lines rather than cross them. (Recall that the World'southward due north magnetic pole is really a south pole in terms of a bar magnet.)

Some incoming charged particles get trapped in the Globe'southward magnetic field, forming ii belts above the atmosphere known as the Van Allen radiation belts subsequently the discoverer James A. Van Allen, an American astrophysicist. (See Figure four.) Particles trapped in these belts form radiation fields (similar to nuclear radiation) so intense that manned space flights avoid them and satellites with sensitive electronics are kept out of them. In the few minutes it took lunar missions to cross the Van Allen radiations belts, astronauts received radiation doses more than twice the allowed annual exposure for radiations workers. Other planets have like belts, peculiarly those having strong magnetic fields like Jupiter.

Figure 4. The Van Allen radiation belts are ii regions in which energetic charged particles are trapped in the Globe'south magnetic field. One belt lies about 300 km above the Earth's surface, the other almost xvi,000 km. Charged particles in these belts migrate along magnetic field lines and are partially reflected abroad from the poles by the stronger fields there. The charged particles that enter the atmosphere are replenished by the Sun and sources in deep outer space.

Back on World, nosotros have devices that employ magnetic fields to contain charged particles. Amidst them are the behemothic particle accelerators that accept been used to explore the substructure of matter. (Run into Figure v.) Magnetic fields not only control the management of the charged particles, they also are used to focus particles into beams and overcome the repulsion of like charges in these beams.

Figure v. The Fermilab facility in Illinois has a big particle accelerator (the most powerful in the earth until 2008) that employs magnetic fields (magnets seen hither in orange) to incorporate and direct its beam. This and other accelerators have been in use for several decades and have allowed us to notice some of the laws underlying all matter. (credit: ammcrim, Flickr)

Thermonuclear fusion (similar that occurring in the Sun) is a hope for a future clean free energy source. One of the most promising devices is the tokamak, which uses magnetic fields to contain (or trap) and direct the reactive charged particles. (Come across Figure 6.) Less exotic, just more immediately practical, amplifiers in microwave ovens use a magnetic field to contain oscillating electrons. These aquiver electrons generate the microwaves sent into the oven.

Figure six. Tokamaks such every bit the i shown in the figure are being studied with the goal of economic production of energy past nuclear fusion. Magnetic fields in the doughnut-shaped device comprise and direct the reactive charged particles. (credit: David Mellis, Flickr)

Mass spectrometers have a variety of designs, and many use magnetic fields to mensurate mass. The curvature of a charged particle'due south path in the field is related to its mass and is measured to obtain mass information. (Run across More Applications of Magnetism.) Historically, such techniques were employed in the get-go direct observations of electron accuse and mass. Today, mass spectrometers (sometimes coupled with gas chromatographs) are used to make up one's mind the make-upward and sequencing of large biological molecules.

Department Summary

• Magnetic force can supply centripetal strength and cause a charged particle to movement in a circular path of radius

  [latex]r=\frac{mv}{qB}\\[/latex]

  where v is the component of the velocity perpendicular to B for a charged particle with mass yard and charge q.

Conceptual Questions

1. How can the movement of a charged particle be used to distinguish between a magnetic and an electric field?

2. High-velocity charged particles can damage biological cells and are a component of radiations exposure in a multifariousness of locations ranging from enquiry facilities to natural background. Draw how y'all could utilise a magnetic field to shield yourself.

3. If a cosmic ray proton approaches the Earth from outer space along a line toward the center of the Earth that lies in the aeroplane of the equator, in what direction will it be deflected by the Earth'due south magnetic field? What nigh an electron? A neutron?

four. What are the signs of the charges on the particles in Figure ix?

Figure 9.

five. Which of the particles in Figure 10 has the greatest velocity, assuming they take identical charges and masses?

Effigy x.

6. Which of the particles in Figure x has the greatest mass, assuming all have identical charges and velocities?

vii. While operating, a high-precision Goggle box monitor is placed on its side during maintenance. The image on the monitor changes color and blurs slightly. Discuss the possible relation of these effects to the Globe'south magnetic field.

Problems & Exercises

If you demand additional back up for these problems, see More Applications of Magnetism.

1. A cosmic ray electron moves at seven.50 × 10⁶m/due south perpendicular to the Globe'due south magnetic field at an altitude where field strength is ane.00 × 10⁻⁵T. What is the radius of the circular path the electron follows?

ii. A proton moves at 7.50 × 10⁷ perpendicular to a magnetic field. The field causes the proton to travel in a circular path of radius 0.800 k. What is the field strength?

3. (a) Viewers of Star Trek hear of an antimatter drive on the Starship Enterprise. One possibility for such a futuristic energy source is to store antimatter charged particles in a vacuum chamber, circulating in a magnetic field, and so excerpt them equally needed. Antimatter annihilates with normal thing, producing pure energy. What strength magnetic field is needed to concur antiprotons, moving at 5.00 × ten^vii chiliad/s in a circular path 2.00 m in radius? Antiprotons have the same mass every bit protons but the reverse (negative) charge. (b) Is this field forcefulness obtainable with today's technology or is it a futuristic possibility?

4. (a) An oxygen-16 ion with a mass of 2.66 × x⁻²⁶kg travels at 5.00 × 10⁶m/s perpendicular to a 1.20-T magnetic field, which makes information technology move in a circular arc with a 0.231-m radius. What positive charge is on the ion? (b) What is the ratio of this accuse to the charge of an electron? (c) Discuss why the ratio found in (b) should be an integer.

five. What radius circular path does an electron travel if information technology moves at the same speed and in the same magnetic field equally the proton in number ii?

6. A velocity selector in a mass spectrometer uses a 0.100-T magnetic field. (a) What electric field force is needed to select a speed of four.00 × ten⁶m/s? (b) What is the voltage between the plates if they are separated by ane.00 cm?

7. An electron in a Idiot box CRT moves with a speed of half dozen.00 × 10⁷m/s, in a direction perpendicular to the Earth's field, which has a force of 5.00 × x⁻⁵T. (a) What strength electric field must be applied perpendicular to the World's field to brand the electron moves in a direct line? (b) If this is done between plates separated past ane.00 cm, what is the voltage applied? (Note that TVs are usually surrounded by a ferromagnetic material to shield against external magnetic fields and avert the need for such a correction.)

8. (a) At what speed volition a proton move in a circular path of the same radius as the electron in question two? (b) What would the radius of the path be if the proton had the same speed as the electron? (c) What would the radius be if the proton had the aforementioned kinetic energy as the electron? (d) The same momentum?

9. A mass spectrometer is being used to separate common oxygen-16 from the much rarer oxygen-18, taken from a sample of old glacial ice. (The relative abundance of these oxygen isotopes is related to climatic temperature at the fourth dimension the ice was deposited.) The ratio of the masses of these two ions is 16 to xviii, the mass of oxygen-xvi is 2.66 × 10⁻²⁶kg, and they are singly charged and travel at five.00 × 10⁶g/s in a one.20-T magnetic field. What is the separation between their paths when they striking a target later on traversing a semicircle?

10. (a) Triply charged uranium-235 and uranium-238 ions are being separated in a mass spectrometer. (The much rarer uranium-235 is used as reactor fuel.) The masses of the ions are iii.90 × 10⁻²⁵kg and iii.95 × 10⁻²⁵kg, respectively, and they travel at 3.00 × 10⁵m/s in a 0.250-T field. What is the separation between their paths when they hit a target after traversing a semicircle? (b) Hash out whether this distance between their paths seems to exist large enough to be practical in the separation of uranium-235 from uranium-238.

Selected Solutions to Problems & Exercises

1. 4.27 m

iii. (a) 0.261 T (b) This strength is definitely obtainable with today's applied science. Magnetic field strengths of 0.500 T are obtainable with permanent magnets.

5. four . 36 × 10 ^{− 4} m

vii. (a) iii.00 kV/m (b) 30.0 V

9. 0.173 yard

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Source: https://courses.lumenlearning.com/physics/chapter/22-5-force-on-a-moving-charge-in-a-magnetic-field-examples-and-applications/

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