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If I say that I am calculating the Lorentz force $F$ applied to a wire carrying a current $i$ at a point $P$ in a magnetic field $B$, would the actual force be opposite of that given by the right hand rule since electrons are actually flowing rather than the positive charge suggested by conventional current (i.e. positive charge doesn't actually flow through a wire)?

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The force on each electron is given by the Lorentz for law which is ,

$$\mathbf{F}=q(\mathbf{v} \times \mathbf{B})$$

Now if we try to apply this to electrons moving in a wire,notice that the direction of electron flow will also be opposite to direction of (conventional)current flow.

So,

\begin{align} \mathbf{F} &=&-e((-\mathbf{v}) \times \mathbf{B})\\&=&e(\mathbf{v} \times \mathbf{B}) \end{align}

which is just the same for positve charges moving in the direction of the conventional current.

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