# Angular momentum associated with a current carrying circular wire

• I
• gurbir_s
In summary, the angular momentum carried by a current carrying circular wire can be calculated by considering the angular momentum of the electrons moving with drift velocity. This can be expressed as L = n m_e v_{drift} r, where r is the radius of the loop and n is the total number of electrons moving in the wire. The magnetic moment of the wire can be seen as the magnetic momentum of the electrons, and can be expressed as mu = gamma N l or mu = gamma L, where N is the total number of electrons, l is the angular momentum for one electron, and L is for all of them.
gurbir_s
How should I calculate the angular momentum carried by a current carrying circular wire? Is it correct to consider the angular momentum of the electrons moving with drift velocity? Like
##L = n m_e v_{drift} r## where ##r## is radius of the loop, and ##n## is total number of electrons moving in the wire?

Can you say more about your setup? You can calculate the forces and torque on such a current-carrying coil in the presence of a B-field, but what do you mean by "angular momentum" in this context? If the coil is free to spin up under the influence of the torque, the angular momentum will increase with time...

Also, is there a commutator in this setup (like with a DC motor)?

There's angular momentum of both the electrons and the electromagnetic field!

berkeman said:
Can you say more about your setup? You can calculate the forces and torque on such a current-carrying coil in the presence of a B-field, but what do you mean by "angular momentum" in this context? If the coil is free to spin up under the influence of the torque, the angular momentum will increase with time...

Also, is there a commutator in this setup (like with a DC motor)?
It is simply a resistive wire in the shape of a circle connected to a DC source.
Actually, I read somewhere that such a loop placed in a magnetic field will simply get aligned in the direction of field. But if it carried an angular momentum, it should have precessed around the applied field.

vanhees71 said:
There's angular momentum of both the electrons and the electromagnetic field!
How to quantify it?

gurbir_s said:
It is simply a resistive wire in the shape of a circle connected to a DC source.
Actually, I read somewhere that such a loop placed in a magnetic field will simply get aligned in the direction of field. But if it carried an angular momentum, it should have precessed around the applied field.
If it carried some angular momentum before the B-field is turned on? I'm still not understanding the question. The loop will experience a torque to try to align it with the B-field, and depending on the damping of the setup, will oscillate for several cycles during that alignment (when there is no commutator).

berkeman said:
If it carried some angular momentum before the B-field is turned on? I'm still not understanding the question. The loop will experience a torque to try to align it with the B-field, and depending on the damping of the setup, will oscillate for several cycles during that alignment (when there is no commutator).
Yes. The angular momentum carried by the circular current carrying wire without applying any field.

Just like in the classical picture, the electrons revolving around nucleus, have an angular momentum and give rise to magnetic moment, by that analogy, since the wire loop has magnetic moment, I want to calculate the angular momentum associated with the current flowing through the wire.

vanhees71 and berkeman
berkeman said:
Ah, so the angular momentum of the electrons in the conduction band moving at the drift current velocity induced by the voltage applied to the coil terminals...

https://en.wikipedia.org/wiki/Drift_velocity
Yes. Sorry, it took me so long to make the question clear.

However, I can't find any reference to the angular momentum in the above mentioned Wikipedia article.

Actually, the magnetic moment of the current loop can be seen as the magnetic momentum of the electrons moving around the loop with the drift velocity. If we consider a loop of radius R, made from wire with cross section S, we can start with the magnetic moment $$\mu=IA$$
and express the current, I, as ##I=nev_d S ## and the area of the loop as ##A=\pi R^2##. Considering that the volume of the wire is ##V_w= 2\pi R S##, we'll have$$\mu=(nev_d a) (\pi R^2) = (2\pi R S) n \frac{e}{2m} (m v_d R) = N \frac{e}{2m}(m v_d R)$$ where N is the total number of electrons, ##N=n V_w## and ##\frac{e}{2m}## is the gyromagnetic factor for the orbital motion of the electrons.
So, it looks like ##\mu=\gamma N l## or ##\mu=\gamma L## where l is the angular momentum for one electron and L is for all of them.

## What is angular momentum in the context of a current-carrying circular wire?

Angular momentum in the context of a current-carrying circular wire refers to the rotational equivalent of linear momentum. It is the quantity of rotation of the wire's current distribution and is a vector quantity, having both magnitude and direction. For a current-carrying circular loop, the angular momentum is associated with the magnetic moment generated by the current.

## How is the angular momentum of a current-carrying circular wire calculated?

The angular momentum (L) of a current-carrying circular wire can be calculated using the formula L = I * A * μ₀ / c, where I is the current, A is the area of the loop, μ₀ is the permeability of free space, and c is the speed of light. This formula arises from the relationship between the magnetic moment and angular momentum in electromagnetism.

## What role does the magnetic moment play in the angular momentum of a current-carrying wire?

The magnetic moment (μ) plays a crucial role in determining the angular momentum of a current-carrying wire. The magnetic moment is given by μ = I * A, where I is the current and A is the area of the loop. The angular momentum is directly proportional to the magnetic moment, highlighting the intrinsic relationship between the two quantities in a current-carrying loop.

## Can the angular momentum of a current-carrying circular wire change, and if so, how?

Yes, the angular momentum of a current-carrying circular wire can change. This can occur if there is a change in the current flowing through the wire, a change in the area of the loop, or if an external torque is applied to the system. Changes in these parameters will alter the magnetic moment and hence the angular momentum of the system.

## What are the practical applications of understanding angular momentum in current-carrying circular wires?

Understanding the angular momentum in current-carrying circular wires has several practical applications, including in the design of electromagnetic devices such as motors, generators, and magnetic storage systems. It is also crucial in understanding the behavior of magnetic materials and in the development of advanced technologies like magnetic resonance imaging (MRI) and particle accelerators.

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