A charge $$Q$$ is moving $$d\vec{l}$$ distance in the magnetic field $$\vec{B}$$. Find the value of work done by $$\vec{B}$$.

The magnetic force on a moving charge is given by $$\vec{F} = Q(\vec{v} \times \vec{B})$$. By the properties of the cross product, this force is always perpendicular to the velocity $$\vec{v}$$ of the charge.

The work done by a force is $$dW = \vec{F} \cdot d\vec{l}$$, where $$d\vec{l}$$ is along the direction of displacement (and hence along $$\vec{v}$$). Since $$\vec{F} \perp \vec{v}$$, the dot product $$\vec{F} \cdot d\vec{l} = 0$$ at every instant.

Therefore, the work done by the magnetic field $$\vec{B}$$ on the charge is zero.