determine the direction of the force on a charge moving in a magnetic field

Force on a Current‑Carrying Conductor

Objective

Determine the direction of the force on a charge moving in a magnetic field and relate it to the force on a current‑carrying conductor.

Magnetic Force on a Moving Charge

A charge \$q\$ moving with velocity \$\vec v\$ in a magnetic field \$\vec B\$ experiences a force

\$ \vec F = q\,\vec v \times \vec B \$

  • The magnitude is \$F = qvB\sin\theta\$, where \$\theta\$ is the angle between \$\vec v\$ and \$\vec B\$.

  • The direction is perpendicular to both \$\vec v\$ and \$\vec B\$ – given by the right‑hand rule.

  • If the charge is negative, the force direction is opposite to that predicted for a positive charge.

Right‑Hand Rule (for positive charge)

  1. Point your fingers in the direction of \$\vec v\$ (velocity).

  2. Curl them toward \$\vec B\$ (magnetic field).

  3. Your thumb points in the direction of \$\vec F\$ (force).

    4. 👉 For a negative charge (e.g., an electron) reverse the thumb direction.

    Force on a Current‑Carrying Conductor

    A conductor of length \$L\$ carrying current \$I\$ in a magnetic field \$\vec B\$ feels a force

    \$ \vec F = I\,\vec L \times \vec B \$

    where \$\vec L\$ points along the direction of conventional current.

    • Magnitude: \$F = BIL\sin\theta\$ (\$\theta\$ = angle between wire and field).

    • Direction: use the same right‑hand rule, but with fingers pointing along the current (instead of charge velocity).

    Fleming’s Left‑Hand Rule (motor rule)

    An alternative for conductors:

    1. Thumb → Motion / Force (\$\vec F\$)

    2. First finger → Magnetic field (\$\vec B\$)

    3. Second finger → Current (\$I\$) (conventional current)

    👉 Align your left hand accordingly; the three fingers are mutually perpendicular.

    Example: Determining Force Direction

    ScenarioGivenForce Direction (Right‑Hand Rule)
    Positive charge moving east (\$\vec v\$) in a field north (\$\vec B\$)\$\vec v\$ = east, \$\vec B\$ = northPoint fingers east, curl north → thumb points up (out of the page)
    Electron (negative) moving west in a field south\$\vec v\$ = west, \$\vec B\$ = southFor a positive charge thumb would point down; reverse for electron → force up
    Wire carrying current north in a field eastCurrent \$I\$ north, \$\vec B\$ eastFingers north, curl east → thumb points up (force out of page)

    Key Points to Remember

    • Force on a moving charge: \$\vec F = q\vec v \times \vec B\$.

    • Force on a wire: \$\vec F = I\vec L \times \vec B\$.

    • Use the right‑hand rule for positive charges/current; reverse for negative charges.

    • Fleming’s left‑hand rule gives the same result for motors (force, field, current).

    • The force is always perpendicular to both the velocity/current and the magnetic field.

    🧲 Understanding these rules lets you predict the motion of charges and the operation of devices like motors and generators. 🚀