Thomson's experiment to find e/m for an electron
If you've jumped to this page directly you may want to read the context of the experiment first.
The two forces on each electron must balance
Since we now know that the particles Thomson was investigating were electrons, that’s what we’ll call them here.
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Each electron has two forces acting on it. The first force arises because charges moving at right angles to a magnetic field experience a force. The other force is due to the electric field caused by the charged plates.
The magnetic force depends on the magnetic field strength, B, the velocity of the electron, v and the charge on the electron, e.
The electric force depends on the charge on the electron and the size of the electric field strength, E.
The magnetic force always acts at right-angles to the direction in which the electron happens to be going at that instant. The magnetic force tends to cause the electron to move in a circle.
The electric force always acts in the same direction, regardless of the direction of motion of the electron. The electric force tends to cause the electron to move in a parabola. You get a parabola when you have constant velocity in one direction and constant acceleration at right angles, like a ball thrown horizontally.
If the electric and magnetic forces are balanced then it doesn’t matter that you would have to combine a parabola with a circle.
Conservation of energy provides a second equation
We can also use conservation of energy to write down another equation saying that the energy lost by the power supply is the same as the energy gained by an electron.
Combining all our equations gives us an expression for e/m mostly in terms of things we can measure like voltages and the plate separation, d.
Finding the value of the magnetic field strength, B
The only sticking point is finding the magnetic field strength, B. Thomson measured it by experiment but if we know enough about the Helmholtz coils we can calculate it.
What was the answer?
If you adjust the electric and magnetic field strengths so the cathode ray particles move in a straight line and then do the sums you should find that the ratio of charge to mass is about 1.8 x 1011 coulombs/kilogram.
Differences with Thomson's actual method
In fact this isn’t quite the same method as Thomson used.
First he used the electric field alone and measured the angle of deflection. Then he used the magnetic field alone and adjusted the field strength until the angle of deflection was the same.
Knowing the angle of deflection he was able to calculate the ratio of charge to mass [mass to charge was what he reported].