# Relation between angular momentum and magnetic moment of an electron (Vector Form)

Why is that minus sign be present in the ratio

my text says -

Mangetic moment (M)= -(e/2m)Lo


where Lo is angular momentum e is the charge on an electron m is the mass of an electron

the reason cited is - the negative sign indicated that the angular momentum and orbital magnetic moment are directed opposite to each other.

however: in some texts

The minus sign is not used

What is the correct variant of the formulae?

## 1 Answer

A minus sign is always there and the difference between the signs in the two texts is due to a different meaning (sign) of $e$. The gyromagnetic ratio is always defined as $$\gamma = \frac{m}{L}$$ where $m$ is the magnetic moment (defined by the contribution to the energy $\Delta E = -\vec m\cdot \vec B$) and $\vec L$ is meant to depict the intrinsic angular momentum (spin) whenever we discuss spinning elementary particles. No minus sign in this equation. However, when you ask how much this constant is for the electron, you obtain $$\gamma = \frac{e}{M_e}$$ where $e$ is charge of the particle and $M_e$ is the mass. However, the charge $e$ in this equation is negative for the electron $$e\lt 0$$ a fact that the PDF you linked to emphasizes between equations 2 and 3. So it may be called $e=q$, at any rate, it's meant to be a negative number.

On the other hand, in the first equation of your question taken from another text, $e$ indicates the positive constant, the elementary charge (the charge of the positron or the proton), so the minus sign has to be written there explicitly to turn it into the charge of the electron.

• What about that statement of sense of direction can you put some light on it ? – Aditya Sher Mar 18 '17 at 16:09
• Aditya, the gyromagnetic ratio is $q/M$ for particles. It means that if the particles have positive charges such as the positron, the magnetic moment and the angular moment have the same direction. If they have a negative charge like the electron, the directions of angular momentum and magnetic moment are oppositely oriented. Here it is assumed that you know that the angular momentum is "equivalent" to some rotational motion that you may obtained with a right hand rule, and so on, and you have some convention for the signs of the magnetic quantities. – Luboš Motl Mar 18 '17 at 16:18
• yes indeed I am aware of the concept angular momentum, and I finally understood this topic @Lubos thanks a ton – Aditya Sher Mar 18 '17 at 16:29