III–4 Application of Magnetic Fields 13.9.2019

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 13.9.2019

Lorenz Force Revisited Let us return to the Lorentz force: F = q[E + (v x B)] and deal with its applications. Let’s start with the magnetic field only. First we show that F = q(v x B)  F = I(L x B) 13.9.2019

Currents are Moving Charges I Let’s have a straight wire 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 13.9.2019

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. 13.9.2019

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 – principle of electro motors. 13.9.2019

Moving Charge in Magnetic Field I Let’s shoot a charged particle q, m by speed v perpendicularly to 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. 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. 13.9.2019

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 = v/(q/m) 1/B r is proportional to the speed over the specific charge. 13.9.2019

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. 13.9.2019

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. 13.9.2019

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. 13.9.2019

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 polarity, only particles with a particular velocity pass the filter. 13.9.2019

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. 13.9.2019

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. 13.9.2019

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 13.9.2019

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 beam goes into magnetic field. 13.9.2019

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. 13.9.2019

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. 13.9.2019

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 13.9.2019

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 13.9.2019

Cyclotrons I Cyclotron is a flat evacuated container which consists of two semi cylindrical parts 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 particles with frequency of their circular motion equal to that of the oscillator will survive. 13.9.2019

Cyclotrons II The radius is given by: r = mv/qB   = v/r = qB/m  f = /2 = qB/2m f is tuned to particular particles. Their final energy depends on how many times they cross the gap. 13.9.2019

Homework Chapter 28 – 1, 2, 5, 14, 21, 23 Due this Wednesday July 31st ! 13.9.2019

Things to read Repeat chapters 27 and 28, excluding 28 7, 8, 9, 10 13.9.2019

The vector or cross product I Let c=a.b Definition (components) The magnitude |c| Is the surface of a parallelepiped made by a,b.

The vector or cross product II 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.)} ^