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).

👉 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

Scenario	Given	Force Direction (Right‑Hand Rule)
Positive charge moving east ($\vec v$) in a field north ($\vec B$)	$\vec v$ = east, $\vec B$ = north	Point fingers east, curl north → thumb points **up** (out of the page)
Electron (negative) moving west in a field south	$\vec v$ = west, $\vec B$ = south	For a positive charge thumb would point **down**; reverse for electron → force **up**
Wire carrying current north in a field east	Current $I$ north, $\vec B$ east	Fingers 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. 🚀