Estimating Magnetic Flux Density using 240 eV Electron Beam and Hall Sensor

Recently I have bought a N42 grade neodymium magnet that boasts function as "salvage" magnets with lifting force equivalent to 600 kg mass on earth. Since my plan for AWAN 3 involves momentum measurement in cosmic ray showers that needs stable magnetic fields, I had to figure out a way to measure the magnetic flux density on the surface of this magnet.

 

The magnet has following physical dimensions:

Magnet diameter: 90 mm

Mount diameter: 120 mm

Thickness: 18 mm

Mount screw: M16

As I did not have any Gauss-meter at hand, I thought of measuring the flux density by cross-checking two different methods:

Method 1: Crude estimation of flux density through bending electron beam of known kinetic energy. 

This is a classic high-school physics demonstration of Lorentz force using specialized cathode ray tubes (CRT) with lots of incorrect assumptions (we'll get to that soon enough), hence "crude estimation". Typically, CRTs uses fluorescent screens to trace the electron beam, the tube I have is one that has partially evacuated glass envelope filled with trace amount of noble gas so it will excite and glow under electron bombardment, making the beam visible. 

In essence, the electron beam inside the CRT is positioned perpendicular to the direction of magnetic field from the magnet. The downward Lorentz force acting on the electrons will "bend" the beam downwards, creating a "vertical displacement". The point is to measure the distance between the magnet to the CRT and see how it varies with the vertical displacement of the beam.

Please take note on all the symbols used.

Variables

vertical displacement, x

magnet to CRT distance, l

Constants

accelerating voltage, V = 240 V 

anode distance to the screen, d = 14 cm 

At l = 100 cm, there is no noticeable deflection on the CRT electron beam so x = 0. I marked the position of the beam on the tube with a Sharpie. Then, by drawing the magnet closer while maintaining perpendicular position as best as I could, I mark down x whenever l changes. x became significant at l = 70 cm, and x became so large that the beam spot touches the edge of the screen at l = 34 cm.

Notice that at l = 30 cm, the electron beam is bent past below the lower edge of the screen. 

Then, all values of x can be converted to radius of curvature, r with the following formula. To save you the trouble of derivation (basic trigonometry really), I'll just provide it here: 

r = sqrt ((x^2)+(d^2)) / 2 cos (90 - arctan (x/d))

Now as I have mentioned, the assumption to make the equation above work is that:

1. The CRT screen must be flat. (reasonable)

2. The electron is bent immediately after it exits anode plate. (not so sure)

3. The field is uniform over the tube so the track would ideally bend into a circle. (very unlikely)

4. It ignores the magnetic properties of glass and the electron gun. 

Then, the radius of curvature, r can be used to compute the field density, B, with the following equation:

B = sqrt (2mV/q(r^2))

assuming non-relativistic electrons. which is true here; where m is the rest mass of electron and q is the elementary charge. B should be in Tesla, and was converted to Gauss by a factor of 10,000. Once all values of B are computed, they are plotted against l. 

Now, any science student would know the field density does not increase linearly to decreasing l. Rightly so because of how the field lines "spreads out" without a ferromagnetic "guide". The relationship fits well to 1/l^2 for short distances though buuuttttt the result is discouraging at best: 

I got less than 1 gauss at a distance of l = 34 cm away from the tube, so I double checked my derivation and computation. Nothing was apparently wrong, which leads me to the second method...

Method 2: Direct measurement of field density with a smartphone hall sensor.   

This one was way easier. Just make the phone coplanar with the magnet without touching (might magnetically fry the phone if it comes too close), then find the position of the sensor inside the phone. This was done with an adjustable stage to find the z-axis position such that it gives the highest reading at fixed distance, l away from the magnet. 

Now the phone's hall sensor and the magnet has been aligned, I measured its field strength in decreasing l just like what I did to the CRT. The phone gives reading in microTeslas, but it is easily converted to Gauss.

When I put the results of these two methods together in the same axes, I was pleasantly surprised that the values of B worked out in Method 1 agrees quite well to what the phone measures, given its uncertainties:

The graph speaks for itself. The x-axis is the displacement of the measuring instrument from the magnet. the value of l.

But with a Hall sensor I can go further and put the phone as close as 10 cm to the magnet. My reading maxxed out close to 50 Gauss, but it is now possible to extrapolate it to 4 cm as this was the distance of the sensitive region of my cloud chamber to the magnet in the initial design of AWAN 3. 

At 4 cm, the magnet gives a pathetic of near 300 Gauss. This is nowhere close to field densities (at least by an order of magnitude!) used by particle hunters early in the 20th century. So. If I want to cut the cost of electricity and buying literally tonnes of precious metals for the solenoid, the only way seems to put the magnet INSIDE the chamber.

That means new blueprints.