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A current carrying conductor is kept stationary in a region of uniform magnetic field, and another conductor, carrying the same amount of current, is moved with a certain velocity in a region of uniform magnetic field, of same magnitude (as the previously mentioned field).

Will the forces on the two be different?

I know that the mathematical formula of magnetic force on a current carrying wire does not account for such inequality, but still are we missing something in this formula ?

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The external force on the whole current-carrying wire in region $V$ can be expressed as

$$ \mathbf F = \int_V \rho\mathbf E_{ext} + \mathbf j \times \mathbf B_{ext}\,d^3\mathbf x. $$ where $\mathbf E_{ext}, \mathbf B_{ext}$ is the external electric and magnetic field and $\rho,\mathbf j$ is the charge and current density.

Stationary and moving wire will have different $\rho,\mathbf j$, so the external force will also be different.

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  • $\begingroup$ Why have you written d^3x , is it that you have written the cross-sectional area of the conductor as d^2x ? $\endgroup$
    – Abhirikshma
    Commented Sep 27, 2015 at 11:23
  • $\begingroup$ It means that the integral is over 3-dimensional space. $\endgroup$
    – Ján Lalinský
    Commented Sep 27, 2015 at 12:00

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