Dimensionless physical quantities

[J-Girl]

Hi:) A question that I don't understand, and my feeble attempt to answer it. Can anybody give any heads up on this one?
If a physical quantity is dimensionless, it has no units attatched to it. Determine if the following constant is dimensionless and show your reasoning"

\alpha= e^2/\hbarc4(Pi)\epsilon_{}_{0}
do all these greek scripted letters just stand for constants, and is it dimensionless because there are no S.I units in the equation? I am assuming constants like 4 and Pi are not units.
im so confused!:(

 

[cepheid]

4 and pi are just numbers, they do not have any physical dimensions.

Yes, the Greek letters are physical constants. You can look them up in order to figure out what their dimensions are. Then you can figure out the dimensions of alpha. If that product of constants is such that all of the units of the individual factors cancel each other out, then alpha is dimensionless.

As an example of dimensionless quantity (albeit one we still assign "units" to): an angle in the radian system is defined as the ratio of two lengths (arc length over radius). As a result, the angle has units of m/m, which cancels out, and the result is unitless (just a number). However, in order to make it clear that this number refers to an angle, we assign it units in "radians", but these are not really units in the traditional sense, since they are measuring a dimensionless quanity, whereas a unit like the metre is used to measure a quantity that has the dimension of length, and a unit like the second is used to measure a quantity that has the dimension of time.

 

[J-Girl]

thanks for ur help:)