Do magnetic forces obey Newton's third law. Verify for two current elements `dvecl_1=dlhati` located at the origin and `dvecl_2=dlhatj` located at (0, R, 0). Both carry current I.

Explanation

Given are two current elements of equal lengths dl where,
`d vecl_(1) = dl hati ` [ located at (0,0,0)]
`d vec l_(2) = dl hat j` [ located at (0,R,0)]
Both the elements carry equal current I.
The magnetic field at location (0, R, 0 ) due to current I flowing in the element `d vecl_(1)` , is given by
`d vecB = mu_(0)/(4 pi) (I dl)/R^(2) hat k`
The magnetic field is directed along `+ Z-`axis according to right hand thumb rule.
As a result of this magnetic field, there will be force exerted on the current carrying element `d vecl_(2)` located at(0, R, 0). The force acting on element `d vecl_(2)` (carrying current I) will be
`vecF_(21) = I (d vecl_(2) xx d vecB)`
` = I [dl hat j xx (mu_(0) Idl)/(4 pi R^(2))hatk]`
` rArr " " vecF_(21) = mu_(0)/(4 pi) (I^(2) dl^(2))/R^(2) hat i" " [ hat j xx hat k = hati]`
So, the force of `d vec l_(2) " due to " d vecl_(1)` will be along `+X-` axis and will have a non-zero value.
The magnetic field produced by the element `d vecl_(2)` at (0, 0, 0) will be zero as the point is lying along the length of the element.
So, there will be no force acting on element `d vecl_(1)` due to element `d vecl_(2)`.
This inequality of forces on the two current elements indicates that magnetic forces do not obey Newton.s third law.