Solutions of Thomas Calculus 11th Edition

Determination of e/m of an electron by using fine beam tube.

Theory:

Electrons are thermally emitted from a surface and accelerated through a potential difference V. The kinetic energy of the accelerated electrons equals the energy they gain as a result of being accelerated through the potential difference. In other words:

½ m v² = eV

and solving for velocity,

v = (2eV/m)^1/2 .

In this equation m is the mass of the electron and e is the charge of the electron.

The beam of electrons enters the region where a magnetic field B is set up by the Helmholz coils. The beam is deflected into a circular path of radius R by the magnetic force and undergoes a centripetal acceleration. This can be expressed as

evB = mv²/r

When the velocity is eliminated between the above two equations, then the charge to mass ratio can be written as

e/m = 2V/(B²r²)

Apparatus:

·         DC power supply

·         Magnetic compass

·         Fine beam tube

·         Magnetic field coils with 150 turns per coil

·         Kepco power supply

·         Digital multimeter

·         Ammeter

·         Voltmeter

Diagram:

Procedure:

Ø  Set the apparatus according to diagram.

Ø  Power up the DC power supply and set acceleration potential U = 200 V. Thermionic emission starts after warming up for a few minutes.

Ø  Add a voltage to the Helmholtz coils using the third knob and look for current I, at which the electron beam is deflected into a closed orbit.

Ø  Move the left slide of the measuring device so that its inner edge, mirror image and escape aperture of the electron beam come to lay on one line of sight. 4.

Ø  Set the right slide for both inside edges to have a distance of 8 cm. 

Ø  Sight the inside edge of the right slide, align it with its mirror image and adjust the coil current I until the electron beam runs tangentially along the slide edge covering the mirror image. The radius of the circle is now r = 4 cm.

Ø  Take the magnetic field B corresponding to this current I.

Ø  Determine the specific charge e/m using the formula  e/m = 2.U_A  / B² . r²   = (*) or the table above (only of U = 200 V and r = 4 cm):

Ø  Compare it to the theoretical value e/m = 0,176*10¹² C/kg!  

Ø  We can measure current ‘i’ with accuracy ±0.005A.

Ø  We can also measure voltage V with accuracy ±0.05V.

Ø  Calculate radius with accuracy ±1mm.

Calculations:

Electron beams crosses the magnetic field at right angle and cause the beam to be deflected by a force Fmeg that is perpendicular to the motion and field. The force act on each electron traveling at speed v

Fmeg = evB

A force which acts on a body moving in a circular path .This force is given by:

F= mV²/r

As Fmag moves perpendicular it is able to provide centripetal force, so

eVB = mV²/r

e/m = /Br

To find v we have

K.E = P.E

1/2mv²= eV

Now from equation

e/m = v/Br

Taking square

e²/m²= 2eV/ B² r² m

e/m= 2V/ B2 r2