Electric vs. Magnetic Forces

Electric and magnetic forces both affect the trajectory of fee particles, yet in qualitatively various ways.

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Learning Objectives

Compare the impacts of the electric and the magnetic fields on the charged particle


Key Takeaways

Key PointsThe force on a fee particle due to an electrical field is directed parallel come the electric field vector in the case of a confident charge, and anti-parallel in the situation of a an adverse charge. It does not rely on the velocity that the particle.In contrast, the magnetic force on a charge fragment is orthogonal to the magnetic field vector, and depends ~ above the velocity that the particle. The right hand ascendancy can be provided to determine the direction that the force.An electrical field might do work-related on a fee particle, while a magnetic ar does no work.The Lorentz force is the combination of the electric and magnetic force, which room often thought about together for valuable applications.Electric field lines are created on confident charges and also terminate on an adverse ones. The ar lines of one isolated fee are directly radially outward. The electrical field is tangent to these lines.Magnetic ar lines, in the instance of a magnet, are created at the north pole and terminate on a south pole. Magnetic poles execute not exist in isolation. Favor in the instance of electric field lines, the magnetic field is tangent come the ar lines. Fee particles will certainly spiral approximately these ar lines.Key Termsorthogonal: Of 2 objects, at appropriate angles; perpendicular to every other.

Electric vs. Magnetic Forces

Force due to both electric and also magnetic pressures will affect the movement of charged particles. However, the resulting adjust to the trajectory of the particles will certainly differ qualitatively between the 2 forces. Below we will quickly review the two varieties of force and also compare and also contrast their effects on a charged particle.

Electrostatic Force and Magnetic pressure on a fee Particle

Recall that in a static, unchanging electric field E the force on a particle with charge q will certainly be:

extF= extqE

Where F is the pressure vector, q is the charge, and E is the electric field vector. Keep in mind that the direction of F is identical to E in the instance of a positivist charge q, and also in the contrary direction in the instance of a negatively charged particle. This electric field might be developed by a larger charge, Q, exhilaration on the smaller sized charge q over a distance r therefore that:

extE=left | frac extF extq ight |= extk left | frac extqQ extqr^2 ight |= extkfracleft extr^2

It must be emphasized that the electric force F plot parallel to the electric field E. The curl of the electrical force is zero, i.e.:

igtriangledown imes extE=0

A an effect of this is the the electrical field may do work and a charge in a pure electrical field will follow the tangent of an electrical field line.

In contrast, recall that the magnetic force on a charged bit is orthogonal come the magnetic ar such that:

extF= extqv imes extB= extqvBsin heta

where B is the magnetic field vector, v is the velocity that the particle and also θ is the angle in between the magnetic field and also the particle velocity. The direction of F can be easily figured out by the use of the right hand rule.


Right Hand Rule: Magnetic areas exert pressures on moving charges. This pressure is among the most straightforward known. The direction of the magnetic force on a moving charge is perpendicular to the airplane formed through v and also B and also follows best hand rule–1 (RHR-1) as shown. The magnitude of the pressure is proportional come q, v, B, and also the sine the the angle in between v and B.


If the fragment velocity happens to be aligned parallel come the magnetic field, or is zero, the magnetic pressure will be zero. This different from the case of an electrical field, wherein the fragment velocity has no bearing, on any given instant, ~ above the magnitude or direction of the electric force.

The angle dependency of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a fragment in an electrical field will move in a straight line follow me an electrical field line.

A further difference in between magnetic and electric pressures is the magnetic fields do no net work, since the particle movement is circular and therefore end up in the very same place. We express this mathematically as:

extW=oint extB cdot extdr = 0

Lorentz Force

The Lorentz pressure is the linked force top top a charged bit due both electric and also magnetic fields, which space often considered together for practical applications. If a fragment of fee q moves with velocity v in the existence of an electrical field E and a magnetic ar B, then it will endure a force:

extF= extq< extE+ extvBsin heta>

Electric and also Magnetic ar Lines

We pointed out briefly above that the motion of charged particles relative to the ar lines differs depending on whether one is handling electric or magnetic fields. There room some remarkable differences between how electric and also magnetic field lines room conceptualized. The electrical field lines from a positive isolated charge are simply a sequence of evenly-spaced, radially command lines pointed outwards indigenous the charge. In the instance of a negative charge, the direction the the field is reversed. The electric field is command tangent to the ar lines. Of course, us imagine the ar lines are more densely packed the bigger the dues are. One can see clearly that the curly of the electric force is zero.


Electric ar Generated by allude Charges: The electric field neighboring three different point charges: (a) A hopeful charge; (b) a negative charge of equal magnitude; (c) a larger negative charge.


If many charges are involved, ar lines are produced on confident charges, and also terminate on an adverse ones.

In the instance of magnets, field lines are created on the phibìc pole (+) and also terminate ~ above the south pole (-) – watch the listed below figure. Magnetic ‘charges’, however, constantly come in bag – there room no magnetic monopoles (isolated phibìc or south poles). The curly of a magnetic field generated by a conventional magnet is therefore constantly non zero. Fee particles will spiral around these ar lines, as long as the particles have some non-zero component of velocity directed perpendicular come the field lines.


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Magnetic Pole Model: The magnetic pole model: 2 opposing poles, north (+) and South (−), be separated by a distance d develop an H-field (lines).


A magnetic field may additionally be produced by a current with the field lines envisioned together concentric circles approximately the current-carrying wire.The magnetic force at any point in this situation can be figured out with the right hand rule, and also will it is in perpendicular come both the current and the magnetic field.


Constant Velocity produce a Straight-Line

If a fee particle’s velocity is parallel to the magnetic field, over there is no network force and also the fragment moves in a straight line.


Learning Objectives

Identify problems required for the bit to relocate in a right line in the magnetic field


Key Takeaways

Key PointsNewton’s first law of movement states that if an object experiences no network force, climate its velocity is constant.A bit with consistent velocity will relocate along a right line v space.If a fee particle’s velocity is completely parallel come the magnetic field, the magnetic field will exert no pressure on the particle and also thus the velocity will continue to be constant.In the situation that the velocity vector is neither parallel nor perpendicular come the magnetic field, the component of the velocity parallel to the field will continue to be constant.Key Termsstraight-line motion: activity that proceeds in a single direction

Constant Velocity to produce Straight-Line Motion

Recall Newton’s very first law that motion. If an item experiences no network force, then its velocity is constant: the object is either at rest (if that velocity is zero), or it moves in a straight line with continuous speed (if that is velocity is nonzero).

There room many instances where a particle may experience no net force. The particle might exist in a vacuum much away from any kind of massive body (that exert gravitational forces) and also electromagnetic fields. Or there can be 2 or more forces ~ above the bit that are well balanced such the the net force is zero. This is the situation for, say, a fragment suspended in an electric field through the electrical force specifically counterbalancing gravity.

If the net force on a bit is zero, climate the acceleration is necessarily zero from Newton’s 2nd law: F=ma. If the acceleration is zero, any velocity the particle has will be kept indefinitely (or till such time together the net pressure is no longer zero). Due to the fact that velocity is a vector, the direction stays unchanged in addition to the speed, therefore the particle continues in a solitary direction, such similar to a right line.

Charged Particles moving Parallel to Magnetic Fields

The pressure a charged bit “feels” because of a magnetic ar is dependency on the angle in between the velocity vector and the magnetic ar vector B . Recall that the magnetic force is:


Zero force When Velocity is Parallel come Magnetic Field: In the case over the magnetic force is zero since the velocity is parallel come the magnetic ar lines.


extF= extqvBsin heta

If the magnetic field and also the velocity room parallel (or antiparallel), then sinθ equals zero and also there is no force. In this situation a charged particle can continue with straight-line motion even in a solid magnetic field. If is in between 0 and also 90 degrees, climate the ingredient of v parallel to B remains unchanged.


Circular Motion

Since the magnetic pressure is constantly perpendicular to the velocity of a charged particle, the particle will undergo circular motion.


Learning Objectives

Describe problems that cause the circular motion of a charged fragment in the magnetic field


Key Takeaways

Key PointsThe magnetic ar does no work, so the kinetic energy and also speed of a charged particle in a magnetic field remain constant.The magnetic force, acting perpendicular to the velocity the the particle, will cause circular motion.The centripetal pressure of the particle is detailed by magnetic Lorentzian pressure so that extqvB=frac extmv^2 extr.Solving for r over yields the gryoradius, or the radius of curvature of the course of a fragment with fee q and also mass m relocating in a magnetic field of toughness B. The gryoradius is then given by extr=frac extmv extqB.The cyclotron frequency (or, equivalently, gyrofrequency) is the number of cycles a particle completes roughly its circular circuit every second and is given by extf=frac extqB2 pi extm.Key Termsgyroradius: The radius of the circular activity of a charged bit in the existence of a uniform magnetic field.cyclotron frequency: The frequency of a fee particle moving perpendicular come the direction of a uniform magnetic ar B (constant magnitude and also direction). Offered by the equality of the centripetal force and also magnetic Lorentz force.

Circular activity of a Charged particle in a Magnetic Field

Magnetic pressures can cause charged corpuscle to relocate in circular or spiral paths. Bit accelerators store protons complying with circular routes with magnetic force. Cosmic rays will certainly follow spiral paths as soon as encountering the magnetic field of astrophysical objects or planets (one instance being Earth’s magnetic field). The balloon chamber photograph in the figure listed below shows fee particles relocating in such bent paths. The curved paths of charged particles in magnetic areas are the basis of a variety of phenomena and can even be offered analytically, such together in a mass spectrometer. Reflects the course traced by corpuscle in a bubble chamber.


Bubble Chamber: Trails of bubbles are developed by high-energy charged particles moving through the superheated liquid hydrogen in this artist’s rendition of a bubble chamber. Over there is a strong magnetic field perpendicular come the page that causes the curved paths of the particles. The radius that the path deserve to be offered to discover the mass, charge, and also energy that the particle.


So, walk the magnetic force reason circular motion? Magnetic force is constantly perpendicular to velocity, so the it walk no job-related on the fee particle. The particle’s kinetic energy and also speed thus remain constant. The direction of activity is affected, but not the speed. This is usual of uniform circular motion. The simplest case occurs when a charged particle moves perpendicular come a uniform B-field, such as displayed in. (If this takes place in a vacuum, the magnetic ar is the leading factor determining the motion. ) Here, the magnetic pressure (Lorentz force) offers the centripetal force


Circular motion of Charged fragment in Magnetic Field: A negatively charged fragment moves in the aircraft of the web page in a an ar where the magnetic ar is perpendicular right into the web page (represented through the tiny circles through x’s—like the tails of arrows). The magnetic pressure is perpendicular to the velocity, and so velocity transforms in direction but not magnitude. Uniform circular movement results.


extF_ extc=frac extmv^2 extr.

Noting that

extsin heta =1

we see that

extF= extqvB.

The Lorentz magnetic force supplies the centripetal force, so these terms room equal:

extqvB=frac extmv^2 extr

solving for r yields

extr=frac extmv extqB

Here, r, dubbed the gyroradius or cyclotron radius, is the radius of curvature of the course of a charged particle with massive m and also charge q, relocating at a speed v perpendicular come a magnetic field of stamin B. In various other words, the is the radius the the circular activity of a charged particle in the visibility of a uniform magnetic field. If the velocity is no perpendicular come the magnetic field, then v is the component of the velocity perpendicular to the field. The component of the velocity parallel come the ar is unaffected, due to the fact that the magnetic force is zero for movement parallel to the field. We’ll check out the results of this case in a later section ~ above spiral motion.

A particle experiencing circular motion because of a uniform magnetic ar is termed to it is in in a cyclotron resonance. The term originates from the name of a cyclic fragment accelerator dubbed a cyclotron, confirmed in. The cyclotron frequency (or, equivalently, gyrofrequency) is the number of cycles a fragment completes around its circular circuit every 2nd and deserve to be found by fixing for v over and substituting in the circulation frequency so that


Cyclotron: A French cyclotron, produced in Zurich, Switzerland in 1937


extf=frac extv2 pi extr

becomes

extf=frac extqB2 pi extm

The cyclotron frequency is trivially offered in radians per second by

omega=frac extqB extm.


Helical Motion

Helical activity results once the velocity vector is no perpendicular come the magnetic ar vector.


Learning Objectives

Describe conditions that lead to the helical activity of a charged bit in the magnetic field


Key Takeaways

Key PointsPreviously, we have seen the circular motion results once the velocity that a charged bit is perpendicular to the magnetic field. The speed and kinetic power of the particle remain constant, however the direction is transformed at each prompt by the perpendicular magnetic force.If the velocity is not perpendicular come the magnetic field, we consider only the component of v the is perpendicular come the field when making our calculations.The ingredient of the velocity parallel to the ar is unaffected, because the magnetic pressure is zero for motion parallel come the field. This to produce helical motion.Charges might spiral along ar lines. If the strength of the magnetic ar increases in the direction that motion, the field will exert a pressure to slow the charges and even reverse their direction. This is well-known as a magnetic mirror.Key Termshelical motion: The activity that is created when one ingredient of the velocity is continuous in magnitude and also direction (i.e., straight-line motion) when the other component is continuous in speed but uniformly varies in direction (i.e., one motion). It is the superposition of straight-line and circular motion.magnetic mirror: A magnetic ar configuration whereby the ar strength changes when moving along a ar line. The mirror effect results in a tendency for fee particles come bounce back from the high ar region.

Helical Motion

In the ar on circular activity we explained the activity of a charged particle with the magnetic field vector to adjust perpendicular come the velocity of the particle. In this case, the magnetic pressure is likewise perpendicular to the velocity (and the magnetic ar vector, the course) at any kind of given moment causing circular motion. The speed and kinetic power of the bit remain constant, but the direction is altered at each instant by the perpendicular magnetic force. Easily reviews this case in the instance of a negative charged bit in a magnetic field directed into the page.


Circular movement of Charged particle in Magnetic Field: A negatively charged bit moves in the plane of the page in a an ar where the magnetic ar is perpendicular into the web page (represented by the small circles v x’s—like the tails that arrows). The magnetic pressure is perpendicular to the velocity, and also so velocity changes in direction yet not magnitude. Uniform circular motion results.


What if the velocity is no perpendicular to the magnetic field? then we consider only the component of v that is perpendicular come the ar when making our calculations, so the the equations of movement become:

extF_ extc=frac extmv_perp^2 extr

extF= extqvBsin heta= extqv_perp extB

The component of the velocity parallel to the ar is unaffected, since the magnetic force is zero for movement parallel come the field. This to produce helical motion (i.e., spiral motion) rather than a circular motion.

shows how electrons not moving perpendicular come magnetic field lines follow the field lines. The component of velocity parallel to the lines is unaffected, and so the charges spiral follow me the ar lines. If field strength boosts in the direction the motion, the field will exert a force to sluggish the charges (and even reverse your direction), creating a kind of magnetic mirror.


Helical Motion and also Magnetic Mirrors: as soon as a charged fragment moves follow me a magnetic field line into a an ar where the ar becomes stronger, the bit experiences a pressure that reduces the ingredient of velocity parallel come the field. This force slows the activity along the field line and also here reverses it, forming a “magnetic mirror. “


The movement of charged particles in magnetic areas are pertained to such various things together the Aurora Borealis or Aurora Australis (northern and southern lights) and also particle accelerators. Charged corpuscle approaching magnetic field lines may obtain trapped in spiral orbits about the lines fairly than crossing them, as checked out above. Part cosmic rays, because that example, follow the Earth’s magnetic field lines, start the setting near the magnetic poles and causing the southern or northern lights v their ionization of molecule in the atmosphere. Those particles that technique middle latitudes should cross magnetic ar lines, and also many room prevented from penetrating the atmosphere. Cosmic rays space a ingredient of elevator radiation; consequently, they provide a greater radiation sheep at the poles 보다 at the equator.


Charged particles Spiral along Earth’s Magnetic field Lines: Energetic electrons and also protons, contents of cosmic rays, native the Sun and also deep outer room often monitor the Earth’s magnetic ar lines rather than overcome them. (Recall that the Earth’s north magnetic pole is really a south pole in regards to a bar magnet. )


Key Takeaways

Key PointsA cyclotron is a kind of fragment accelerator in which charged particles accelerate outwards native the center along a spiral path. The corpuscle are held to a spiral trajectory by a revolution magnetic ar and increased by a promptly varying electrical field.The cavity magnetron is a high-powered vacuum tube that generates microwaves using the communication of a currently of electrons through a magnetic field. The magnetron has applications in radar, heating, and also lighting.Mass spectrometers measure up the mass-to-charge ratio of fee particles through the use of electromagnetic areas to segregate particles with various masses and/or charges. It have the right to be provided to recognize the element composition of a molecule or sample.Key Termscyclotron: an early particle accelerator in which charged particles were produced at a main source and increased spirally exterior through a addressed magnetic and alternate electric fields.mass spectrometer: A device used in fixed spectrometry to find the mass composition of a provided substance.magnetron: A machine in which electrons room made to resonate in a specially shaped chamber and thus develop microwave radiation; offered in radar, and also in microwave ovens.

Examples and Applications – motion of a Charged fragment in a Magnetic Field

Overview

Recall the the charged particles in a magnetic field will monitor a one or spiral path relying on the alignment of your velocity vector with the magnetic ar vector. The results of such motion deserve to have profoundly handy applications. Countless technologies are based upon the movement of charged particles in electromagnetic fields. Us will explore some of these, consisting of the cyclotron and also synchrotron, cavity magnetron, and also mass spectrometer.

Cyclotrons and also Synchrotrons

A cyclotron is a kind of fragment accelerator in which fee particles accelerate outwards native the facility along a spiral path. The corpuscle are organized to a spiral trajectory by a static magnetic ar and sped up by a swiftly varying (radio frequency ) electric field.


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Cyclotron Sketch: map out of a bit being increased in a cyclotron, and also being ejected with a beamline.


Cyclotrons accelerate charged fragment beams utilizing a high frequency alternative voltage i m sorry is used between two “D”-shaped electrodes (also dubbed “dees”). Secondary static magnetic field is applied in perpendicular direction to the electrode plane, permitting particles to re-encounter the accelerating voltage plenty of times in ~ the exact same phase. To achieve this, the voltage frequency must complement the particle’s cyclotron resonance frequency,

extf=frac extqB2 pi extm

with the relativistic massive m and its charge q. This frequency is provided by equality the centripetal force and also magnetic Lorentz force. The particles, injected near the facility of the magnetic field, rise their kinetic energy only when recirculating with the gap between the electrodes; thus they take trip outwards along a spiral path. Your radius will rise until the corpuscle hit a target in ~ the perimeter the the vacuum chamber, or leave the cyclotron using a beam tube, permitting their use. The particles increased by the cyclotron can be offered in particle therapy come treat some species of cancer. Additionally, cyclotrons room a great source of high-energy beams because that nuclear physics experiments.

A synchrotron is an innovation upon the cyclotron in i m sorry the guiding magnetic field (bending the particles into a closeup of the door path) is time-dependent, being synchronized to a particle beam of boosting kinetic energy. The synchrotron is just one of the first accelerator ideas that allow the construction of large facilities, due to the fact that bending, beam focusing and also acceleration have the right to be separated into different components.

Cavity Magnetron

The cavity magnetron is a high-powered vacuum tube that generates microwaves using the communication of a stream of electrons through a magnetic field. Every cavity magnetrons consist of a hot cathode with a high (continuous or pulsed) an adverse potential produced by a high-voltage, direct-current strength supply. The cathode is constructed into the center of an evacuated, lobed, circular chamber. A magnetic field parallel to the filament is applied by a irreversible magnet. The magnetic field causes the electrons, attractive to the (relatively) optimistic outer component of the chamber, to spiral external in a circular path, a repercussion of the Lorentz force. Spaced approximately the rim of the chamber are cylindrical cavities. The caries are open along your length and also connect the common cavity space. Together electrons sweep past these openings, lock induce a resonant, high-frequency radio field in the cavity, i m sorry in turn reasons the electrons to bunch right into groups.


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Cavity Magnetron Diagram: A cross-sectional chart of a resonant cavity magnetron. Magnetic currently of force are parallel to the geometric axis of this structure.


The sizes of the cavities determine the resonant frequency, and thereby the frequency of emitted microwaves. The magnetron is a self-oscillating machine requiring no external elements other than a strength supply. The magnetron has actually practical applications in radar, heating (as the main component the a microwave oven), and lighting.

Mass Spectrometry

Mass spectrometry is one analytical method that measures the mass-to-charge proportion of charged particles. The is provided for determining masses that particles and also determining the elemental composition of a sample or molecule.

Mass analyzers different the ion according to your mass-to-charge ratio. The adhering to two laws govern the dynamics of charged particles in electric and also magnetic fields in a vacuum:

extF= extQ( extE+ extv imes extB) (Lorentz force)

extF= extma

Equating the above expressions because that the force used to the ion yields:

( extm/ extQ) exta= extE+ extv imes extB

This differential equation in addition to initial conditions completely determines the activity of a charged fragment in terms of m/Q. There are many species of fixed analyzers, utilizing either static or dynamic fields, and magnetic or electrical fields, however all run according come the above differential equation.

The following number illustrates one kind of fixed spectrometer. The deflections of the particles space dependent top top the mass-to-charge ratio. In the instance of isotopic carbon dioxide, every molecule has actually the very same charge, yet different masses. The fixed spectrometer will certainly segregate the particles spatially enabling a detector to measure the mass-to-charge ratio of every particle. Since the fee is known, the absolute mass can be figured out trivially. The loved one abundances have the right to be inferred indigenous counting the number of particles of each offered mass.

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Mass Spectrometry: Schematics of a simple mass spectrometer with sector type mass analyzer. This one is because that the measurement of carbon dioxide isotope ratios (IRMS) as in the carbon-13 urea breath test.