Do electron spin like a ball?

you have heard that electron have spin and magnitude of that spin is $\dfrac{1}{2}$.

well it's actually $\dfrac{1}{2}\hbar$ we just take $\hbar = 1$ 

and we all know that $\hbar$ is reduced plank constant.

but is electron really spinning like ball and having angular momentum of $\dfrac{1}{2}\hbar$ ?

hmmm... what is velocity of the electron then?

let's try to calculate that , let's assume electron is solid sphere and it's radius is $r_e = 2.81×10^{-15}$ meters.

(check this https://en.m.wikipedia.org/wiki/Classical_electron_radius)

from high-school physics you can calculate moment of inertia of the solid sphere. and it's turn out to be $I = \dfrac{2}{5}mr^2$.

and angular velocity $\omega = \dfrac{v}{r}$ 

so we have all the equipment.

now we know that angular momentum $L = I\omega$ 

also $L = \dfrac{1}{2}\hbar$ 

so we have

$I\omega = \dfrac{1}{2}\hbar$

putting Value of $I$ and $\omega$

$\dfrac{2}{5}mr^{2}\dfrac{v}{r} = \dfrac{1}{2}\hbar$

we want velocity so 

$v = \dfrac{5\hbar}{4mr}$

value of $\hbar$ is $1.055×10^{-34}$ and value of $r$ is $2.81×10^{-15}$

putting this value we get 

$v = 5.15×10^{10} \frac{m}{s}$

which is clearly greater than speed of light.

so electron don't have spin of $\frac{1}{2}$?

no, they do but they aren't spinning like a ball, that is our model is wrong. 'Spin' is intrinsic form of angular momentum unlike orbital angular momentum.

So thinking that electron are spinning like ball leads to contradiction to relativity.

maybe in future posts we'll see what intrinsic means.