## Electric vs. Magnetic Forces

Electric and magnetic forces both affect the trajectory of charged particles, but in qualitatively different ways.

You are watching: Like kinds of magnetic poles repel while unlike kinds of magnetic poles

### Learning Objectives

Compare the results of the electric and the magnetic fields on the fee particle

### Key Takeaways

Key PointsThe pressure on a fee particle due to an electric field is command parallel come the electrical field vector in the situation of a confident charge, and anti-parallel in the situation of a an adverse charge. It does not depend on the velocity that the particle.In contrast, the magnetic pressure on a charge bit is orthogonal come the magnetic ar vector, and also depends ~ above the velocity that the particle. The right hand rule can be offered to identify the direction that the force.An electric field might do work-related on a fee particle, while a magnetic field does no work.The Lorentz force is the mix of the electric and also magnetic force, which space often taken into consideration together for practical applications.Electric ar lines are created on optimistic charges and also terminate on an adverse ones. The ar lines of an isolated fee are straight radially outward. The electric field is tangent to these lines.Magnetic field lines, in the case of a magnet, are generated at the phibìc pole and also terminate top top a south pole. Magnetic poles carry out not exist in isolation. Like in the situation of electrical field lines, the magnetic field is tangent to the ar lines. Fee particles will spiral about these field lines.Key Termsorthogonal: Of two objects, at best angles; perpendicular to each other.

### Electric vs. Magnetic Forces

Force as result of both electric and also magnetic pressures will affect the motion of fee particles. However, the resulting change to the trajectory that the particles will differ qualitatively between the 2 forces. Below we will quickly review the two varieties of force and also compare and also contrast their impacts on a fee particle.

### Electrostatic Force and Magnetic force on a charged Particle

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

extF= extqE

Where F is the pressure vector, q is the charge, and E is the electrical field vector. Keep in mind that the direction of F is identical to E in the case of a positivist fee q, and in the contrary direction in the case of a negatively charged particle. This electrical field might be established by a bigger charge, Q, acting on the smaller sized charge q end a street r so that:

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

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

igtriangledown imes extE=0

A consequence of this is that 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 the 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 ar vector, v is the velocity that the particle and θ is the angle in between the magnetic field and also the fragment velocity. The direction the F can be easily figured out by the usage of the ideal hand rule.

Right Hand Rule: Magnetic areas exert forces on relocating charges. This force is one of the most simple known. The direction that the magnetic pressure on a relocating charge is perpendicular to the aircraft formed by v and also B and also follows appropriate hand rule–1 (RHR-1) together shown. The magnitude of the force is proportional to q, v, B, and also the sine that the angle in between v and B.

If the fragment velocity happens to it is in aligned parallel come the magnetic field, or is zero, the magnetic pressure will it is in zero. This different from the instance of an electrical field, where the bit velocity has no bearing, on any type of given instant, on the magnitude or direction the the electric force.

The angle dependency of the magnetic field additionally causes charged particles to move perpendicular come the magnetic field lines in a one or helical fashion, while a particle in an electric field will relocate in a right line follow me an electrical field line.

A additional difference in between magnetic and also electric pressures is that magnetic areas do no net work, since the particle activity is circular and therefore ends up in the exact same place. Us express this mathematically as:

extW=oint extB cdot extdr = 0

### Lorentz Force

The Lorentz pressure is the an unified force on a charged particle due both electric and magnetic fields, which space often taken into consideration together for valuable applications. If a bit of charge q moves v velocity v in the existence of an electric field E and also a magnetic field B, climate it will experience a force:

extF= extq< extE+ extvBsin heta>

### Electric and also Magnetic ar Lines

We discussed briefly over that the activity of charged particles relative to the field lines differs depending on whether one is dealing with electric or magnetic fields. There room some noteworthy differences in between how electric and also magnetic ar lines are conceptualized. The electric field lines native a optimistic isolated fee are merely a succession 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 directed tangent to the field lines. Of course, we imagine the ar lines are more densely packed the bigger the fees are. One deserve to see plainly 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 an unfavorable charge of equal magnitude; (c) a larger an unfavorable charge.

If lot of charges are involved, ar lines are produced on hopeful charges, and terminate on an unfavorable ones.

In the instance of magnets, field lines are generated on the north pole (+) and also terminate ~ above the south pole (-) – see the listed below figure. Magnetic ‘charges’, however, always come in pairs – there space no magnetic monopoles (isolated phibìc or south poles). The curl of a magnetic ar generated by a traditional magnet is therefore constantly non zero. Fee particles will spiral roughly these ar lines, as lengthy as the particles have actually some non-zero component of velocity directed perpendicular to the ar lines.

Magnetic Pole Model: The magnetic pole model: two opposing poles, phibìc (+) and also South (−), be separated by a street d create an H-field (lines).

A magnetic ar may likewise be created by a existing with the ar lines envisioned together concentric circles roughly the current-carrying wire.The magnetic pressure at any point in this instance can be established with the appropriate hand rule, and also will be perpendicular to both the current and the magnetic field.

## Constant Velocity to produce a Straight-Line

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

### Learning Objectives

Identify problems required because that the fragment to move in a straight line in the magnetic field

### Key Takeaways

Key PointsNewton’s first law of motion states that if an item experiences no net force, then its velocity is constant.A bit with continuous velocity will move along a directly line with 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 remain constant.In the situation that the velocity vector is no parallel nor perpendicular to the magnetic field, the component of the velocity parallel to the field will stay constant.Key Termsstraight-line motion: movement that proceeds in a solitary direction

### Constant Velocity produces Straight-Line Motion

Recall Newton’s an initial law that motion. If an object experiences no net force, climate its velocity is constant: the thing is one of two people at remainder (if that velocity is zero), or it move in a straight line with constant speed (if the velocity is nonzero).

There are many instances where a particle might experience no net force. The particle might exist in a vacuum far away from any kind of massive body (that exert gravitational forces) and electromagnetic fields. Or there can be two or more forces top top the bit that are well balanced such that the net pressure is zero. This is the instance for, say, a particle suspended in an electrical field with the electrical force exactly counterbalancing gravity.

If the net pressure on a bit is zero, climate the acceleration is have to zero from Newton’s 2nd law: F=ma. If the acceleration is zero, any velocity the particle has will be maintained indefinitely (or until such time as the net pressure is no longer zero). Since velocity is a vector, the direction stays unchanged along with the speed, therefore the particle proceeds in a solitary direction, such similar to a right line.

### Charged Particles moving Parallel come Magnetic Fields

The pressure a charged particle “feels” because of a magnetic field is dependency on the angle between the velocity vector and also the magnetic ar vector B . Recall the the magnetic pressure is:

Zero pressure When Velocity is Parallel to Magnetic Field: In the case over the magnetic pressure is zero due to the fact that the velocity is parallel come the magnetic ar lines.

extF= extqvBsin heta

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

## Circular Motion

Since the magnetic force is always perpendicular come the velocity the a fee particle, the particle will undergo circular motion.

### Learning Objectives

Describe conditions that cause the circular motion of a charged bit in the magnetic field

### Key Takeaways

Key PointsThe magnetic ar does no work, therefore the kinetic energy and also speed the a charged particle in a magnetic field remain constant.The magnetic force, exhilaration perpendicular come the velocity the the particle, will reason circular motion.The centripetal pressure of the fragment is detailed by magnetic Lorentzian force so that extqvB=frac extmv^2 extr.Solving for r over yields the gryoradius, or the radius of curvature the the route of a fragment with charge q and also mass m moving in a magnetic field of stamin 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 about its circular circuit every 2nd and is given by extf=frac extqB2 pi extm.Key Termsgyroradius: The radius that the circular motion of a charged particle in the existence of a uniform magnetic field.cyclotron frequency: The frequency the a charged particle relocating perpendicular come the direction the a uniform magnetic field B (constant magnitude and direction). Offered by the equality the the centripetal force and also magnetic Lorentz force.

### Circular motion of a Charged fragment in a Magnetic Field

Magnetic pressures can cause charged particles to move in one or spiral paths. Bit accelerators store protons complying with circular paths with magnetic force. Cosmic rays will follow spiral paths when encountering the magnetic field of astrophysical objects or planets (one instance being Earth’s magnetic field). The balloon chamber picture in the figure listed below shows charged particles relocating in such bent paths. The curved yellowcomic.com of charged particles in magnetic fields are the communication of a number of phenomena and can also be provided analytically, such as in a massive spectrometer. Reflects the path traced by corpuscle in a bubble chamber.

Bubble Chamber: Trails of bubbles are developed by high-energy charged particles relocating through the superheated liquid hydrogen in this artist’s rendition the a bubble chamber. Over there is a solid magnetic ar perpendicular come the web page that causes the curved paths of the particles. The radius of the path can be supplied to uncover the mass, charge, and energy the the particle.

So, go the magnetic force reason circular motion? Magnetic pressure is constantly perpendicular to velocity, so the it go no job-related on the charged particle. The particle’s kinetic energy and speed therefore remain constant. The direction of activity is affected, yet not the speed. This is common of uniform one motion. The simplest situation occurs as soon as a charged bit moves perpendicular to a uniform B-field, such as presented in. (If this takes place in a vacuum, the magnetic ar is the dominant factor determining the motion. ) Here, the magnetic force (Lorentz force) offers the centripetal force

Circular motion of Charged bit in Magnetic Field: A negatively charged fragment moves in the plane of the page in a an ar where the magnetic field is perpendicular into the web page (represented by the small circles v x’s—like the tails of arrows). The magnetic pressure is perpendicular come the velocity, and so velocity changes in direction but not magnitude. Uniform circular activity 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, referred to as the gyroradius or cyclotron radius, is the radius that curvature of the path of a charged fragment with mass m and also charge q, moving at a speed v perpendicular come a magnetic field of stamin B. In other words, that is the radius of the circular activity of a charged particle in the visibility of a uniform magnetic field. If the velocity is not perpendicular come the magnetic field, then v is the ingredient of the velocity perpendicular come the field. The ingredient of the velocity parallel to the ar is unaffected, due to the fact that the magnetic pressure is zero for activity parallel come the field. We’ll explore the after-effects of this instance in a later on section on spiral motion.

A fragment experiencing one motion as result of a uniform magnetic ar is termed to be in a cyclotron resonance. The term originates from the surname of a cyclic particle accelerator dubbed a cyclotron, confirmed in. The cyclotron frequency (or, equivalently, gyrofrequency) is the variety of cycles a particle completes approximately its circular circuit every 2nd and can be discovered by solving for v above and substituting in the circulation frequency for this reason 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 given in radians per 2nd by

omega=frac extqB extm.

## Helical Motion

Helical motion results when the velocity vector is not perpendicular come the magnetic field vector.

### Learning Objectives

Describe problems that result in the helical movement of a charged particle in the magnetic field

### Key Takeaways

Key PointsPreviously, we have seen the circular activity results when the velocity the a charged bit is perpendicular come the magnetic field. The speed and also kinetic energy of the fragment 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 think about only the component of v the is perpendicular come the field when making our calculations.The component of the velocity parallel come the ar is unaffected, due to the fact that the magnetic force is zero for motion parallel come the field. This produce helical motion.Charges might spiral along field lines. If the strength of the magnetic field increases in the direction that motion, the field will exert a force to slow-moving the charges and also even reverse their direction. This is known as a magnetic mirror.Key Termshelical motion: The movement that is produced when one component of the velocity is continuous in magnitude and direction (i.e., straight-line motion) when the various other component is continuous in speed but uniformly different in direction (i.e., one motion). That is the superposition that straight-line and circular motion.magnetic mirror: A magnetic field configuration where the field strength alters when moving along a field line. The mirror effect results in a tendency for fee particles come bounce ago from the high field region.

### Helical Motion

In the ar on circular movement we defined the motion of a charged fragment with the magnetic ar vector to adjust perpendicular come the velocity of the particle. In this case, the magnetic pressure is additionally perpendicular to the velocity (and the magnetic field vector, the course) at any given moment bring about circular motion. The speed and also kinetic energy of the particle remain constant, but the direction is changed at each immediate by the perpendicular magnetic force. Quickly reviews this instance in the situation of a negative charged fragment in a magnetic ar directed right into the page.

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

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

extF_ extc=frac extmv_perp^2 extr

extF= extqvBsin heta= extqv_perp extB

The ingredient of the velocity parallel to the ar is unaffected, since the magnetic pressure is zero for activity parallel to the field. This produces helical motion (i.e., spiral motion) rather than a one motion.

shows exactly how electrons not relocating perpendicular to magnetic field lines follow the field lines. The component of velocity parallel come the lines is unaffected, and so the fees spiral along the field lines. If field strength increases in the direction the motion, the field will exert a force to sluggish the dues (and also reverse your direction), creating a sort of magnetic mirror.

Helical Motion and Magnetic Mirrors: as soon as a charged particle moves along a magnetic field line into a an ar where the field becomes stronger, the fragment experiences a pressure that to reduce the component of velocity parallel to the field. This force slows the motion along the ar line and also here reverses it, forming a “magnetic mirror. “

The activity of charged particles in magnetic areas are pertained to such various things together the Aurora Borealis or Aurora Australis (northern and southern lights) and particle accelerators. Charged particles approaching magnetic ar lines may gain trapped in spiral orbits about the lines fairly than crossing them, as checked out above. Some cosmic rays, for example, follow the Earth’s magnetic ar lines, beginning the atmosphere near the magnetic poles and also causing the southern or northern lights with their ionization of molecules in the atmosphere. Those corpuscle that technique middle latitudes should cross magnetic ar lines, and many are prevented indigenous penetrating the atmosphere. Cosmic rays space a component 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 protons, materials of cosmic rays, native the Sun and also deep outer space often follow the Earth’s magnetic field lines fairly than overcome them. (Recall the the Earth’s north magnetic pole is yes, really a southern pole in regards to a bar magnet. )

### Key Takeaways

Key PointsA cyclotron is a type of bit accelerator in which charged particles advice outwards from the facility along a spiral path. The particles are organized to a spiral trajectory by a revolution magnetic ar and increased by a swiftly varying electric field.The cavity magnetron is a high-powered vacuum tube that generates microwaves utilizing the communication of a present of electrons with a magnetic field. The magnetron has actually applications in radar, heating, and also lighting.Mass spectrometers measure up the mass-to-charge ratio of charged particles v the usage of electromagnetic areas to segregate particles with various masses and/or charges. It deserve to be provided to determine the elemental composition of a molecule or sample.Key Termscyclotron: an early particle accelerator in which charged particles were produced at a central source and increased spirally exterior through a solved magnetic and alternative electric fields.mass spectrometer: A an equipment used in mass spectrometry to uncover the mass ingredient of a provided substance.magnetron: A an equipment in i beg your pardon electrons are made to resonate in a specially shaped chamber and thus create microwave radiation; used in radar, and in microwave ovens.

### Overview

Recall the the fee particles in a magnetic ar will follow a circular or spiral path depending on the alignment of your velocity vector v the magnetic ar vector. The aftermath of such motion have the right to have profoundly valuable applications. Many technologies are based upon the movement of charged particles in electromagnetic fields. Us will explore some the these, consisting of the cyclotron and synchrotron, cavity magnetron, and mass spectrometer.

### Cyclotrons and Synchrotrons

A cyclotron is a kind of bit accelerator in which charged particles advice outwards from 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 ) electrical field.

Cyclotron Sketch: map out of a bit being accelerated in a cyclotron, and being ejected v a beamline.

Cyclotrons advice charged bit beams making use of a high frequency alternative voltage which is applied between 2 “D”-shaped electrodes (also referred to as “dees”). An additional static magnetic field is applied in perpendicular direction come the electrode plane, permitting particles come re-encounter the speeding up voltage many times in ~ the exact same phase. To accomplish 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 magnetic Lorentz force. The particles, injected near the facility of the magnetic field, rise their kinetic power only when recirculating with the gap in between the electrodes; thus they travel outwards along a spiral path. Their radius will increase until the particles hit a target at the perimeter that the vacuum chamber, or leaving the cyclotron utilizing a beam tube, enabling their use. The particles increased by the cyclotron have the right to be used in fragment therapy to treat some varieties of cancer. Additionally, cyclotrons are a great source that high-energy beams for nuclear physics experiments.

A synchrotron is an advancement upon the cyclotron in i m sorry the guiding magnetic field (bending the particles into a close up door path) is time-dependent, gift synchronized come a fragment beam of increasing kinetic energy. The synchrotron is among the an initial accelerator concepts that enable the construction of large facilities, due to the fact that bending, beam focusing and acceleration deserve to be be separated into various components.

### Cavity Magnetron

The cavity magnetron is a high-powered vacuum tube the generates microwaves making use of the interaction of a present of electrons v a magnetic field. Every cavity magnetrons covers a warm cathode with a high (continuous or pulsed) an adverse potential created by a high-voltage, direct-current strength supply. The cathode is constructed into the center of an evacuated, lobed, circular chamber. A magnetic ar parallel to the filament is imposed by a permanent magnet. The magnetic field reasons the electrons, attractive to the (relatively) optimistic outer component of the chamber, to spiral external in a circular path, a consequence of the Lorentz force. Spaced about the pickled in salt of the chamber space cylindrical cavities. The cavities are open along their length and also connect the typical cavity space. Together electrons sweep previous these openings, lock induce a resonant, high-frequency radio field in the cavity, i beg your pardon in turn causes the electron to bunch into groups.

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

The size of the cavities recognize the resonant frequency, and thereby the frequency that emitted microwaves. The magnetron is a self-oscillating device requiring no external facets other than a strength supply. The magnetron has practical applications in radar, heater (as the main component that a microwave oven), and lighting.

### Mass Spectrometry

Mass spectrometry is one analytical an approach that actions the mass-to-charge ratio of fee particles. That is provided for identify masses of particles and determining the element composition of a sample or molecule.

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

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

extF= extma

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

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

This differential equation together with initial conditions fully determines the motion of a charged bit in terms of m/Q. There room many varieties of fixed analyzers, utilizing either static or dynamic fields, and magnetic or electrical fields, however all run according to the above differential equation.

The following figure illustrates one kind of massive spectrometer. The deflections that the particles room dependent top top the mass-to-charge ratio. In the instance of isotopic carbon dioxide, each molecule has actually the very same charge, but different masses. The mass spectrometer will segregate the particles spatially allowing a detector to measure the mass-to-charge ratio of every particle. Due to the fact that the charge is known, the absolute mass deserve to be identified trivially. The relative abundances deserve to be inferred indigenous counting the variety of particles the each offered mass.

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Mass Spectrometry: Schematics of a straightforward mass spectrometer v sector type mass analyzer. This one is for the measure of carbon dioxide isotope ratios (IRMS) together in the carbon-13 urea breath test.