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In class, we talked about the discovery of electrons. The book stated, "..results all gave same value for charge to mass ratio. This showed that there was some (particle) common to all materials in a discharge tube" Is this true that particles must necessarily have different values for $\frac{e}{m}$? Is there a law for this?

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No, there's not, but given that there's no particular reason for two particle species to have the same ratio, and that it could be any real number at all, it would be quite the coincidence for two different particles to have the same ratio.

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Numerically, it may be found that two particles are having same e/m ratio. e.g. electron and its antiparticle would have same e/m ratio. The only difference is in its sign.

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