Write the expression for Lorentz magnetic force on a particle of charge 'q' moving with velocity 1 in a magnetic field B. Show that no work is done by this force on the charged particle ?

The Lorentz magnetic force is a fundamental concept in electromagnetism that describes the force experienced by a charged particle moving through a magnetic field. To express this mathematically, we can use the formula for the Lorentz force, which combines both electric and magnetic components. In your case, since we are focusing on the magnetic force when a particle of charge 'q' moves with a given velocity 'v' in a magnetic field 'B', we can simplify our focus to the magnetic part of the Lorentz force.

Expression for Lorentz Magnetic Force

The magnetic force (${\mathbf{F}}_B$) acting on a charged particle can be expressed as:

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

In this equation:

• ${\mathbf{F}}_B$ is the magnetic force vector.
• q is the charge of the particle.
• $\mathbf{v}$ is the velocity vector of the charged particle.
• $\mathbf{B}$ is the magnetic field vector.
• $\times$ denotes the cross product, which indicates that the magnetic force is perpendicular to both the velocity of the particle and the magnetic field.

Understanding the Nature of Work Done

To show that no work is done by the magnetic force on the charged particle, we need to consider the definition of work in physics. Work ($W$) is defined as the dot product of force and displacement:

$W=\mathbf{F}\cdot \mathbf{d}$

For the magnetic force, we can substitute:

$W={\mathbf{F}}_B\cdot \mathbf{d}$

Since ${\mathbf{F}}_B$ is perpendicular to $\mathbf{v}$ (and thus also to the displacement $\mathbf{d}$), the angle between the force vector and the displacement vector is 90 degrees. In this case, the dot product simplifies to:

$W=|{\mathbf{F}}_B||\mathbf{d}|\cos ({90}^{\circ })=0$

Illustrative Example

Imagine a charged particle, like an electron, moving in a circular path within a magnetic field. The magnetic force acts as a centripetal force, keeping the electron in motion but not doing any work on it, because the direction of the force is always changing to keep the particle moving in a curve. It does not increase or decrease the kinetic energy of the particle, which is a key indicator of work being done.

Final Remarks

In summary, the Lorentz magnetic force acting on a charged particle moving in a magnetic field can be expressed as ${\mathbf{F}}_B=q\mathbf{v}\times \mathbf{B}$. Importantly, because this force is always perpendicular to the particle's velocity, it does no work on the particle, maintaining its kinetic energy constant while altering its direction of motion. This principle plays a critical role in various applications, such as in the design of particle accelerators and the functioning of electric motors.