# Lectures 12-13 (Ch. 27) Magnetism, Magnetic Force 1. Introduction (permanent magnets, magnetic field, B) 2. Magnetic force. 3. Motion of q in B (Uniform.

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Lectures 12-13 (Ch. 27) Magnetism, Magnetic Force 1. Introduction (permanent magnets, magnetic field, B) 2. Magnetic force. 3. Motion of q in B (Uniform B, Nonuniform B) 4. Motion of q in crossed B and E 5. Force on a segment of a current 6. Torque on a loop of a current 7. Applications

Permanent magnets Similar to electric attraction and/or repulsion. But each magnet has two opposite poles like an electric dipole has two opposite charges. The poles can not be separated.

Magnets produce magnetic field, B The structure of B of a magnet is similar to the structure of E of a dipole. But B lines are always closed lines (contrary to E lines which start on + and end on - q) Again: magnetic monopoles do not exist

Electric currents also produce B Amper’s equivalence hypothesis Magnet consists of the loops of the current. They are due to circulation of electrons around nucleus. I

The core is very hot so that iron is not magnetic. B is attributed to dynamo effect of the circulating electric current in the core of the earth. The rotation of the Earth plays a part in generating the currents. Evidence for 171 magnetic field reversals during the past 71 million years has been reported. Spin axis

Magnetic force on a moving charge 1.Work done by magnetic force=0 2. Force does not change the magnitude of v, just the direction

Right hand rule (RHR) Let v vector enter a palm of your right hand. Then curl the fingers to grab B vector. Your thumb points in direction of F vector for positive charge and opposite to F vector for negative charge.

Units of B [B]=[F]/[q][v]=Ns/Cm=N/Am=1T (tesla) 1T=10 4 G (gauss) Range of B: Magnetically shielded room 10 -10 G Interstellar space 10 -6 G Earth's magnetic field 0.5 G Small bar magnet 100 G Strong lab magnet 10 Tesla Big electromagnet 10 Tesla Surface of neutron star 10 8 T Magstar 10 11 T

The largest superconducting magnet used in LHC

Motion of the charged particles in B Circular motion (in plane ) Cyclotron frequency If B is out of page +q makes clockwise rotation (hertz) (radian per sec)

Cyclotron is a device to accelerate the charged particles B v F Top view: “Dees” (two metal half- cylindrical empty cans). Cyclotron frequency does not depend on v ! Charged particles are accelerated by E (t) in the interval between the “Dees”. E(t) is a periodic function with the frequency = cyclotron frequency. When v approaches c, relativistic effects should be taken into account: E(t) E One has to tune the frequency of the applied E(t) to provide the further acceleration of the particles. Synchrophasotron.

Example. You need to built a cyclotron to accelerate the protons from K 0 =1MeV to K f =29MeV. You may provide B=1T and maximum voltage, V=10kV. 1)What should be the frequency of the applied voltage? 2)What should be the minimum radius of “Dees”? 3) How much time does it require?

Discovery of the positron in a bubble chamber, in the strong B Prediction of a positron: Paul Dirac, 1928 Discovery of a positron: Carl Anderson, 1932 Paul Dirac 1902–1984

Uniform B, V has a component in B direction Circular motion in a plane perpendicular to B with the radius is combined with a motion with constant velocity, along B, resulting in a helical path. Pitch (the distance between two neighboring turns):

Example1. At t=0 proton with v=(1.5x10 5 i+2x10 5 j)m/s enters at the origin of the coordinate system the region with B=0.5iT. Describe and plot the pass. Find the coordinates of the proton at t=T/2 where T is the period of the circle. F y z y z B VyVy F R Example2. A proton enters the region of the uniform magnetic field, B=B x iT, experiencing the magnetic force F=F z kN. What could you tell about initial velocity of the proton?

Nonuniform B Magnetic bottle: the force is toward the center, the particles are trapped. Radius is the largest in the center where B is the weakest (R~1/B). Most of the particles reaching the ends slow down and turn back under the action of force, oscillating between the ends. The most energetic particles may escape only on the most ends where B is uniform.

The Van-Allen Radiation Belts protect life on the earth from the cosmyc radiation Aurora borealis (Northern lights) The principle of magnetic bottle: charged particles are trapped by the nonuniform B of the Earth. They may escape only near the poles regions resulting in Northern and Southern lights.

Tokamak (toroidal magnetic coil ) Plasma confinement for nuclear fusion Positively charged ions and negatively charged electrons in a fusion plasma are at very high temperatures (~10 6 K), and have correspondingly large velocities. In order to maintain the fusion process, particles from the hot plasma must be confined in the central region, or the plasma will rapidly cool. Magnetic confinement fusion devices exploit the fact that charged particles in a magnetic field feel a magnetic force and follow helical paths along the field lines.ions electrons magnetic force

Magnetic deflection If charged particle starts to orbit around B lines. If the size of the region with B is smaller then the radius of the orbit, the particle will be deflected at the exit of the region from original direction by a distance (see figure): Example. A particle with q=2.15 μC and m=3.2x10 -11 kg enters the region of B as shown on Fig. Find its displacement at the exit of the region and a total displacement on the wall. =0.42T

Motion of q in B ┴ E. Velocity selector Independent on the mass and q Particles with this velocity will be undeflected. Particles with larger velocity will be deflected by B. Particles with smaller velocity will be deflected by E.

Mass spectrometer (discovery of the isotopes by J.J. Tomson,1913) (1) (2) It consists of two regions (1) and (2). In the region (1) there is crossed E and B. It is a velocity selector. Particles propagating with v=E/B are undeflected and enter the region (2). In the region (2) only B’ is present ( it may be the same as in region (1) or different from it). Particles start to move in circles and after making a half of circle they strike the photographic plate. Measuring the diameter of the circle allows to calculate the mass of the particle if q is known. Using this device isotopes were discovered. Neon 20 and 22

Example. In the mass spectrometer, a distance between P and P’ plates is 0.5cm, B=B’=0.7T. The particles propagate undeflected at v=2.68x10 4 m/s. 1. Find voltage between the plates. (2) Find the distance, d, between the lines on the photographic plate for singly charged positive ions of 24Mg and 26Mg. 1amu=1.66x10 -27 kg.

The cathod ray tube

J.J. Thomson e/m experiment, 1897 discovery of e Sir J.J.Tomson (1856 -1940) Nobel prize,1906 V

The Hall effect (Edwin Hall, 1879) When current flows along the strip conductor in the presence of B perpendicular to the strip the deflection of carriers results in inducing of voltage across the strip (Hall voltage ). Separation of charges occurs till qE= qvB→ V H =EL=vBL 1.Sign of V H allows one to define the sign of careers (see figure) 2.Measuring of V H allows to find out drift velocity : 3. Measuring of V H and I allows to find out n (density of carriers): L

The Hall resistance: 1T10T RHRH B Quantum Hall effect, Nobel Prize 1982, von Klitzing Fractional Hall effect, Nobel Prize 1998,Stromer,Tsui,Laughlin

Force on a segment of a current Example. x y z

Example. Electromagnetic rail gun (a simple motor) Data: I=1000A, B=1T,m=50kg,L=1m Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth (11.2km.s) x x F

Example Force on a half of a circle A quarter of a circle I = - F=0

Loudspeaker

Magnetic dipole moment (A is the area of the loop)

Torque on a current loop Ƭ

Back to Amper’s equivalence = SN μμ Analogy between electric and dipole moments S N μ B SN μ Stable equilibrium μ tends to be parallel to B E - + p S p + - E B Stable equilibrium p tends to be parallel to E x x

Solenoid (or magnetic coil) many loops of current with the same direction of current

DC current motor

Force on a current loop in a nonuniform B NS Replace a loop with equivalent magnet NS attraction repulsion Replace a loop with equivalent magnet

Attraction of an unmagnetized iron to the magnet NS B Unmagnetized iron:μ=0 1. Alignment (randomly oriented μ become parallel to B, i.e. iron become magnetized ) 2. Attraction(opposite poles attract each other) SN B SN

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