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

Force on a Current‑Carrying Conductor

Learning Objective

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

1. Magnetic Force on a Single Charge

The magnetic force on a charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$ is given by the Lorentz force law:

$$\mathbf{F} = q\,\mathbf{v}\times\mathbf{B}$$

Key points:

• The force is perpendicular to both $\mathbf{v}$ and $\mathbf{B}$.
• The magnitude is $F = qvB\sin\theta$, where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{B}$.
• The direction is found using the right‑hand rule for the cross product.

2. Right‑Hand Rule for $\mathbf{v}\times\mathbf{B}$

To apply the rule:

1. Point the fingers of your right hand in the direction of $\mathbf{v}$ (the motion of a positive charge).
2. Sweep the fingers toward the direction of $\mathbf{B}$.
3. Thumb points in the direction of the force $\mathbf{F}$ on a positive charge.
4. For a negative charge, the force is opposite to the thumb direction.

3. From a Single Charge to a Current‑Carrying Conductor

A conductor carrying a current $I$ contains many charge carriers (usually electrons) moving with drift velocity $\mathbf{v}_d$. The total magnetic force on a length $L$ of the conductor is the sum of the forces on all carriers:

$$\mathbf{F} = I\,\mathbf{L}\times\mathbf{B}$$

where $\mathbf{L}$ is a vector of magnitude $L$ directed along the conventional current (positive to negative).

4. Direction of the Force on a Conductor

Use the same right‑hand rule, but replace $\mathbf{v}$ with the direction of the current $\mathbf{I}$ (or $\mathbf{L}$):

1. Point the fingers in the direction of the conventional current (from positive to negative).
2. Rotate the hand so the fingers can sweep toward the magnetic field direction $\mathbf{B}$.
3. The thumb then points in the direction of the force on the conductor.

5. Summary Table of Directions

Configuration	Current Direction $\mathbf{I}$	Magnetic Field $\mathbf{B}$	Resulting Force $\mathbf{F}$
Case 1	Into the page (×)	To the right (+x)	Upwards (+y)
Case 2	To the right (+x)	Upwards (+y)	Into the page (×)
Case 3	Upwards (+y)	Into the page (×)	To the left (–x)

6. Example Problem

Problem: A straight wire of length $0.30\ \text{m}$ carries a current of $5.0\ \text{A}$ to the north. It is placed in a uniform magnetic field of $0.20\ \text{T}$ directed eastwards. Find the magnitude and direction of the magnetic force on the wire.

Solution:

1. Identify vectors:
   • $\mathbf{L}$ points north (positive $y$).
   • $\mathbf{B}$ points east (positive $x$).
2. Use $F = I L B \sin\theta$. Here $\theta = 90^\circ$, so $\sin\theta = 1$.
3. Calculate magnitude: $$F = (5.0\ \text{A})(0.30\ \text{m})(0.20\ \text{T}) = 0.30\ \text{N}$$
4. Direction: Apply right‑hand rule.
   • Thumb (current) points north.
   • Fingers point east (field).
   • Thumb of the right hand (force) points upward, i.e., out of the page.

7. Common Misconceptions

• Electron motion vs. conventional current: The right‑hand rule uses conventional current direction. If you consider electron drift, the force direction is opposite.
• Angle $\theta$: The force is zero when $\mathbf{v}$ (or $\mathbf{I}$) is parallel or antiparallel to $\mathbf{B}$ because $\sin 0^\circ = 0$.
• Force on the wire vs. force on individual charges: The net force on the wire is the sum of forces on all carriers, giving the simple $I\mathbf{L}\times\mathbf{B}$ expression.

8. Practice Questions

1. A rectangular loop of wire carries a current $I$ clockwise when viewed from above. The loop lies in a uniform magnetic field directed into the page. Determine the direction of the net force on each side of the loop.
2. A proton moves with speed $2.0\times10^6\ \text{m s}^{-1}$ perpendicular to a magnetic field of $0.5\ \text{T}$. Calculate the magnitude of the magnetic force if the proton’s charge is $+e$.
3. Explain why a current‑carrying wire placed parallel to a magnetic field experiences no magnetic force.

9. Key Take‑aways

• The magnetic force on a moving charge is given by $\mathbf{F}=q\mathbf{v}\times\mathbf{B}$.
• For a current‑carrying conductor, the force is $\mathbf{F}=I\mathbf{L}\times\mathbf{B}$.
• Direction is found using the right‑hand rule with conventional current direction.
• The force is maximal when $\mathbf{v}$ (or $\mathbf{I}$) is perpendicular to $\mathbf{B}$ and zero when they are parallel.