Derive an expression for magnetic force $\vec{F}$ acting on a straight conductor of length $L$ carrying current I in an external magnetic field $\vec{B}$. Is it valid when the conductor is in zig-zag form? Justify.

(a) Write the expression for the Lorentz force on a particle of charge $$q$$ moving with a velocity $$\vec{v}$$ in a magnetic field $$\vec{B}$$. When is the magnitude of this force maximum? Show that no work is done by this force on the particle during its motion from a point $$\vec{r_1} \text { to point } \vec{r}_2 \text {. }$$

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(b) A long straight wire $$A B$$ carries a current I. A particle (mass $$m$$ and charge $$q$$ ) moves with a velocity $$\vec{v}$$, parallel to the wire, at a distance $$d$$ from it as shown in the figure. Obtain the expression for the force experienced by the particle and mention its directions.