Magnetic field due to a long straight current-carrying wire

Around a long straight wire, the magnetic field lines are concentric circles centered on the wire.

The direction of magnetic field is given by the right-hand thumb rule.

Magnetic force on a moving charge

The magnetic force on a charge is $\vec{F} = q(\vec{v} \times \vec{B})$

Here, the charge is negative, so the force direction is opposite to $\vec{v} \times \vec{B}$.

Given condition

The force on the charge is parallel to the direction of current in the wire.

So we need a velocity direction such that for a negative charge, the force comes out along the current.

Check possible motions

Let the current in the wire be upward. Then the magnetic field circles the wire.

Take the charge at a point to the right of the wire. At that point, the magnetic field is into the page.

• If the charge moves away from the wire, its velocity is toward the right.
• Then $\vec{v} \times \vec{B} = (\text{right}) \times (\text{into page}) = \text{upward}$
• Since the charge is negative, the actual force is opposite to this, i.e. downward.

This is opposite to the current, so A is incorrect.

• If the charge moves towards the wire, its velocity is toward the left.
• Then $\vec{v} \times \vec{B} = (\text{left}) \times (\text{into page}) = \text{downward}$
• For a negative charge, force is opposite, i.e. upward.

This is parallel to the current. So B is correct.

• If the charge moves parallel to the wire, then $\vec{v}$ is along the wire and $\vec{B}$ is circular around it, so force will be radial (towards or away from wire), not along the current.

Hence C and D are incorrect.

Conclusion

The charge must move towards the wire.