Force on a current carrying conductor

[Physicslearner500039]

TL;DR

Force on a current carrying conductor

I am studying the theory of force on a current carrying conductor in a uniform magnetic field, the Force is $F=ILB$. I am slightly confused here let us say the conductor is placed in the uniform magnetic field $B$, then if the current is passing through the conductor then that current would also produce a magnetic field say $B1$. Will these two fields B and B1 not interact with each other? Please advise.

 

[etotheipi]

As a simple explanation, the wire will not interact with its own magnetic field, but will only interact with the external magnetic field. So when calculating the force on the wire, you only need to take into account the external magnetic field. At points in space around the wire, the resultant magnetic field can be obtained by the vector sum of the magnetic fields due to the wire and the external source (and you could measure it by, for example, using a 'test charge' of arbitrarily small charge).

 

[Physicslearner500039]

But in the textbook i am referring to it refers to only the external magnetic field but not the resultant magnetic field. Is it because the field of the current carrying conductor is negligible? May be i have to go through more thoroughly. Sorry i understood now.

 

[etotheipi]

But in the textbook i am referring to it refers to only the external magnetic field but not the resultant magnetic field. Is it because the field of the current carrying conductor is negligible? May be i have to go through more thoroughly.

As I mentioned, the wire will only interact with the external magnetic field. If $\vec{B}$ is the external field, then the force on a small charge element is $d\vec{F} = d^3 x \rho \vec{v} \times \vec{B}$, which means that$$\vec{F} = \int d^3 x \rho \vec{v} \times \vec{B} = \int d^3 x \vec{J} \times \vec{B}$$The force per unit length is then $\vec{f} = \vec{I} \times \vec{B}$

 

[Physicslearner500039]

Yes I have understood now, the distinction i have to make is force on the wire, force near the wire. The magnetic fields are not the same and hence the forces.

 

[etotheipi]

Yes I have understood now, the distinction i have to make is force on the wire, force near the wire. The magnetic fields are not the same and hence the forces.

Remember that you only get a magnetic or electric force when a field interacts with a charge, so it doesn't make sense to say 'force near the wire'. What you can say is that the field in the vicinity of the wire is obtained via the superposition (vector sum) of the magnetic fields from both sources.

I think the magnetic field due to the wire, at the position of any point on the wire, is undefined in any case.

 

[dlgoff]

Here's a good reference IMO.

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/forwir2.html