Force on a Moving Charge in a Magnetic Field

Understanding the force experienced by a charged particle moving through a magnetic field is fundamental to electromagnetism. This force, known as the Lorentz force, plays a crucial role in various phenomena and applications, from particle accelerators to electric motors.

The Lorentz Force Formula

A moving charge in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field.

The magnitude of this force depends on the charge, velocity, magnetic field strength, and the angle between velocity and field. The direction is determined by the right-hand rule.

The Lorentz force ( $\vec{F}$ ) on a charge ( $q$ ) moving with velocity ( $\vec{v}$ ) in a magnetic field ( $\vec{B}$ ) is given by the vector cross product: $\vec{F} = q(\vec{v} imes \vec{B})$ .

The magnitude of the force is $F = |q|vB\sin heta$ , where $heta$ is the angle between $\vec{v}$ and $\vec{B}$ .

Key points:

If the charge is zero, the force is zero.
If the velocity is zero, the force is zero.
If the velocity is parallel or antiparallel to the magnetic field ( $heta = 0^\circ$ or $180^\circ$ ), $\sin heta = 0$ , so the force is zero.
The force is always perpendicular to both $\vec{v}$ and $\vec{B}$ .

Direction of the Force: The Right-Hand Rule

To determine the direction of the force on a positive charge, use the right-hand rule: Point the fingers of your right hand in the direction of the velocity ( $\vec{v}$ ). Curl your fingers towards the direction of the magnetic field ( $\vec{B}$ ). Your thumb will then point in the direction of the force ( $\vec{F}$ ). For a negative charge, the force is in the opposite direction.

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This directional aspect is crucial. Since the force is always perpendicular to the velocity, it does not do work on the charge. This means the kinetic energy of the charge remains constant, and its speed does not change. The magnetic force only changes the direction of the velocity.

Consequences of the Magnetic Force

When a charged particle moves perpendicular to a uniform magnetic field, the force acts as a centripetal force, causing the particle to move in a circular path. The radius of this circular path can be determined by equating the magnetic force to the centripetal force: $|q|vB = \frac{mv^2}{r}$ , which leads to $r = \frac{mv}{|q|B}$ .

What happens to the kinetic energy of a charged particle moving in a magnetic field?

The kinetic energy remains constant because the magnetic force is always perpendicular to the velocity and does no work.

Applications and Examples

This principle is the basis for many technologies, including:

Mass Spectrometers: Used to separate ions based on their mass-to-charge ratio.
Cyclotrons and Synchrotrons: Particle accelerators that use magnetic fields to guide charged particles.
Cathode Ray Tubes (CRTs): Used in older televisions and monitors, where magnetic fields steer electron beams.
Electric Motors: The interaction between magnetic fields and current-carrying wires (which are essentially moving charges) generates torque.

Remember: The magnetic force only affects moving charges. Stationary charges are unaffected by magnetic fields.

JEE Physics Mastery: Key Takeaways

Concept	Formula	Direction	Work Done	Effect on KE
Force on Moving Charge	$\vec{F} = q(\vec{v} \times \vec{B})$	Right-Hand Rule	Zero	Constant
Magnitude	$F = \|q\|vB\sin\theta$	N/A	N/A	N/A
Circular Motion Radius	$r = \frac{mv}{\|q\|B}$	N/A	N/A	N/A

Learning Resources

Force on a Moving Charge in a Magnetic Field - Physics Classroom(documentation)

Provides a clear explanation of the magnetic force on moving charges, including the right-hand rule and the formula.

Lorentz Force - Wikipedia(wikipedia)

A comprehensive overview of the Lorentz force, its history, mathematical formulation, and applications.

Magnetic Force on a Moving Charge - Khan Academy(video)

A video tutorial explaining the concept of magnetic force on moving charges and demonstrating the right-hand rule.

JEE Physics: Magnetism - Force on Moving Charge(video)

A YouTube video specifically tailored for JEE preparation, covering the force on a moving charge in a magnetic field.

Understanding the Lorentz Force Law - Physics LibreTexts(documentation)

An in-depth explanation of the Lorentz force law, its components, and its implications in classical mechanics.

Magnetic Field and Force - MIT OpenCourseware(documentation)

Lecture notes from MIT covering magnetic fields and forces, including detailed derivations and examples.

Force on a Charged Particle in a Magnetic Field - BYJU'S(blog)

Explains the force on a charged particle in a magnetic field with examples and diagrams relevant to competitive exams.

JEE Physics: Motion of Charged Particle in Magnetic Field(video)

A video focusing on the motion of charged particles in magnetic fields, including circular and helical paths, crucial for JEE.

The Lorentz Force - A Detailed Explanation(documentation)

An interactive explanation of the Lorentz force with animations and detailed mathematical treatments.

JEE Main 2024 Physics: Magnetism and Matter - Force on Moving Charge(blog)

A resource specifically designed for JEE Main preparation, covering the force on moving charges within the magnetism chapter.

Force on a Moving Charge in Magnetic Field