Electric Deflection

Lab manual pp. 47-49

Overview:
Our goal in this lab is to measure the deflection of electrons in an electric field.  We will use the equations of motion to solve the equation of the path of an electron.  We also want to obtain the value
    alpha = the effective length of a capacitor / the actual length
for the given cathode ray tube (CRT).

Notes:
You should remember that charges in an electric field experience a force of magnitude: F = q*E.  We want to find the motion of an electron in an electric field.  We can do this using Newton's Second Law: F = m*a.  The acceleration of a charge in an electric field E is, therefore:  a = (q*E)/m
Furthermore, if the force is constant, we can use our equations of motion from Phys 151 or 170.  Let me remind you what they are:
y = v_oy * t + (1/2) * a_y * t^2
and
v = v_ox + a_x * t
In most cases, the electric field (and therefore the force) is not constant.  For example, a point charge has an electric field which changes with radial distance (r).  But for one special case, the field is nearly constant.  That case is the capacitor.  The field inside a parallel-plate capacitor is nearly constant between the plates, and outside the plates, the field is nearly zero.


the ideal parallel plate capacitor

That means that the electron is accelerated upwards while it is between the plates, and it's path, assuming it had an initial horizontal velocity, will be parabolic.  As soon as it leaves the capacitor, E = 0, so the force on the electron is zero, and the electron will move in a linear path.  Often we do not know the value of E, but we do know the value of V, the voltage between the plates.  When the electric field is constant, the potential difference is simply:  V_d = E * t.  Rearranging, this is:
E = (V_d) / t.
It is simple to calculate, given the equations above, that the vertical deflection (or the vertical distance that the electron travels) is:

vertical deflection = d = (e*(V_d) * l ) * ( l /2 + L ) / (t * m * (v_0)^2)
where L is the distance the electron travels outside of the capacitor.  This can be simplified further if we know that the electron was accelerated by a horizontal electric field, and the energy given by this field is e * V_g, where V_g is the potential difference.  The kinetic energy of the electron as it enters the capacitor is: 1/2 * m * (v_0) ^ 2 = e * V_g
Our equation, then is:
2*d = (V_d / V_g) * (l / t) * (l /2 + L)
In a real capacitor, the field is not constant.  The fields outside the capacitor wrap around the capacitor.  The effect of this is that the electron experiences some extra deflection due to these fringing fields.  We can model this behavior by imagining our ideal capacitor lengthened by some extra length.  The length is: l + extra length = l_eff.  We plug this value into our equation and we get:
2*d =  (V_d / V_g) * ( l _eff/ t) * ( l_eff/2 + L - extra/2)
(notice that L is shortened by the extra length)
simplified, we get:
2*d =  (V_d / V_g) * ( l_eff/ t) * ( l /2 + L )
This is of the form y = m*x + b, where b = 0, y = 2*d, and x = V_d.

Procedure:
This apparatus uses high voltage.  You must follow these steps EXACTLY or risk injury.  Any one who does not follow the safety rules which I have proscribed will be dismissed and will receive a zero for this lab.
Safety Rule 1:  BEFORE CONNECTING OR DISCONNECTING ANY WIRES MAKE SURE THE POWER SUPPLY IS TURNED OFF.
Safety Rule 2: NEVER TOUCH THE METAL CONNECTORS ONCE THE POWER HAS BEEN TURNED ON.

1) Measure the DC acceleration voltage, V_g.  To measure the DC voltage on the power supply:

  • Use the banana connectors to connect the multimeter to the power supply as shown below.
  • Turn the multimeter dial to DCV, at the 1500V setting.
  • Turn on the power supply.  Once on, DO NOT TOUCH the connections.
  • Read the voltage.
  • Turn off the power supply before unhooking the connectors.
    • 2) Measure the amplitude of the AC deflection voltage, V_d.  Follow the above steps except you should put your red connector in the AC socket on the power supply, and turn your multimeter to ACV, at the 200V setting.  Do not remove the connectors once you are finished reading the voltage.
    3) Using a plastic ruler, measure the length line on the CRT representing 2*d.
    4) Change the voltage of the AC deflection voltage by turning the deflection knob on the power supply.  Read the voltage on the multimeter (which is still connected) and measure the new 2*d.
    5) Make a table of five voltages and their corresponding 2*d.
    6) After reading five voltages, turn off the power supply.
    7) Make a graph of 2*d versus V_d and find the slope.
    8) Calculate alpha = L_eff/L