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Why does a current carrying conductor parallel to the magnetic field not experience a force?

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The Lorentz force law says that the force $\mathbf{F}$ due to a magnetic field $\mathbf{B}$ on a particle with charge $q$ moving with velocity $\mathbf{v}$ is $$ \mathbf{F} = q\,\mathbf{v}\times\mathbf{B}. \tag{1} $$ If $\mathbf{v}$ is parallel to $\mathbf{B}$, then $\mathbf{v}\times\mathbf{B} \propto \mathbf{B}\times \mathbf{B}= 0$ (because the cross-product of any vector with itself is zero), so the force is zero.

In a wire carrying a current in the direction $\mathbf{u}$, the charge carriers are moving with velocity $\mathbf{v}\propto\mathbf{u}$, so we again have $\mathbf{v}\times\mathbf{B}=0$ if $\mathbf{u}$ is parallel to $\mathbf{B}$.

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