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III–4 Application of Magnetic Fields

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Main Topics

• Applications of Lorentz Force• Currents are Moving Charges• Moving Charges in El. & Mag.• Specific charge Measurements• The Story of the Electron.• The Mass Spectroscopy.• The Hall Effect.• Accelerators

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Lorenz Force Revisited

• Let us return to the Lorentz force:

and deal with its applications.

• Let’s start with the magnetic field only. First, we show that

)]([ BvEqF

)()( BLIFBvqF

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Currents are Moving Charges I

• Let’s have a straight wire with the length L perpendicular to magnetic field and charge q, moving with speed v in it.

• Time it takes charge to pass L is: t = L/v

• The current is: I = q/t = qv/L q = IL/v

• Let’s substitute for q into Lorentz equation:

F = qvB = ILvB/v = ILB

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Currents are Moving Charges II

• If we want to know how a certain conductor in which current flows behaves in magnetic field, we can imagine that positive charges are moving in it in the direction of the current. Usually, we don’t have to care what polarity the free charge carriers really are.

• We can illustrate it on a conductive rod on rails.

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Currents are Moving Charges III

• Let’s connect a power source to two rails which are in a plane perpendicular to the magnetic field. And let’s lay two rods, one with positive free charge carriers and the other with negative ones.

• We see that since the charges move in the opposite directions and the force on the negative one must be multiplied by –1, both forces have the same direction and both rods would move in the same direction . This is a principle of electro motors.

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Moving Charge in Magnetic Field I

• Let’s shoot a charged particle q, m by speed v perpendicularly to the field lines of homogeneous magnetic field of the induction B.

• The magnitude of the force is F = qvB and we can find its direction since FvB must be a right-turning system. Caution negative q changes the orientation of the force!

• Since F is perpendicular to v it will change permanently only the direction of the movement and the result is circular motion of the particle.

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Moving Charge in Magnetic Field II

• The result is similar to planetary motion. The Lorentz force must act as the central or centripetal force of the circular movement:

mv2/r = qvB• Usually r is measured to identify particles:

• r is proportional to the speed and indirectly proportional to the specific charge and magnetic induction.

B

vr

mq

1

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Moving Charge in Magnetic Field III

• This is basis for identification of particles for instance in bubble chamber in particle physics.• We can immediately distinguish polarity.

• If two particles are identical than the one with larger r has larger speed and energy.

• If speed is the same, the particle with larger specific charge has smaller r.

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Specific Charge Measurement I

• This principle can be used to measure specific charge of the electron.

• We get free electrons from hot electrode (cathode), then we accelerate them forcing them to path across voltage V, then let them fly perpendicularly into the magnetic field B and measure the radius of their path r.

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Specific Charge Measurement II

• From: mv2/r = qvB v = rqB/m• This we substitute into equation describing

conservation of energy during the acceleration:

• mv2/2 = qV q/m = 2V/(rB)2 • Quantities on the right can be measured. B

is calculated from the current and geometry of the magnets, usually Helmholtz coils.

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Specific Charge of Electron I

• Originally J. J. Thompson used different approach in 1897.

• He used a device now known as a velocity filter.• If magnetic field B and electric field E

are applied perpendicularly and in a right direction, only particles with a particular velocity v pass the filter.

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Specific Charge of Electron II

• If a particle is to pass the filter the magnetic and electric forces must compensate:

qE = qvB v = E/B

• This doesn’t depend neither on the mass nor on the charge of the particle.

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Specific Charge of Electron III

• So what exactly did Thompson do? He:• used an electron gun, now known as CRT.

• applied zero fields and marked the undeflected beam spot.

• applied electric field E and marked the deflection y.

• applied also magnetic field B and adjusted its magnitude so the beam was again undeflected.

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Specific Charge of Electron IV

• If a particle with speed v and mass m flies perpendicularly into electric field of intensity E, it does parabolic movement and its deflection after a length L:

y = EqL2/2mv2

• We can substitute for v = E/B and get:

m/q = L2B2/2yE

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Mass Spectroscopy I

• The above principles are also the basis of an important analytical method mass spectroscopy. Which works as follows:• The analyzed sample is ionized or separated e.g. by GC

and ionized.• Then ions are accelerated and run through a velocity

filter.• Finally the ion beam goes perpendicularly into

magnetic field and number of ions v.s. radius r is measured.

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Mass Spectroscopy II

• The number of ions as a function of specific charge is measured and on its basis the chemical composition can be, at least in principle, reconstructed.

• Modern mass spectroscopes usually modify fields so the r is constant and ions fall into one aperture of a very sensitive detector.

• But the basic principle is anyway the same.

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The Hall Effect I

• Let’s insert a thin, long and flat plate of material into uniform magnetic field. The field lines should be perpendicular to the plane.

• When current flows along the long direction a voltage across appears.

• Its polarity depends on the polarity of free charge carriers and its magnitude caries information on their mobility.

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The Hall Effect II

• The sides of the sample start to charge until a field is reached which balances the electric and magnetic forces:

qE = qvdB

• If the short dimension is L the voltage is:

Vh = EL = vdBL

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Accelerators

• Accelerators are built to provide charged particles of high energy. Combination of electric field to accelerate and magnetic field to focus (spiral movement) or confine the particle beam in particular geometry.• Cyclotrons

• Synchrotrons

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Cyclotrons I

• Cyclotron is a flat evacuated container which consists of two semi cylindrical parts (Dees) with a gap between them. Both parts are connected to an oscillator which switches polarity at a certain frequency.

• Particles are accelerated when they pass through the gap in right time. The mechanism serves as an frequency selector. Only those of them with frequency of their circular motion equal to that of the oscillator will survive.

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Cyclotrons II

• The radius is given by:

• r = mv/qB = v/r = qB/m • f = /2 = qB/2m

• f is tuned to particular particles. Their final energy depends on how many times they cross the gap. Limits: size Ek ~ r2, relativity

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Homework

• Chapter 28 – 1, 2, 5, 14, 21, 23

• Due this Wednesday July 30

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Things to read

• Repeat chapters 27 and 28,

excluding 28 - 7, 8, 9, 10

• Advance reading 28 – 7, 8, 9, 10

The vector or cross product I Let c=a.b

Definition (components)

The magnitude |c|

kjijki bac

sinbac

Is the surface of a parallelepiped made by a,b.

The vector or cross product II

zyx

zyx

zyx

bbb

aaa

uuu

c

The vector c is perpendicular to the plane made by the vectors a and b and they have to form a right-turning system.

ijk = {1 (even permutation), -1 (odd), 0 (eq.)}

^