Write an expression for the Lorentz force on a charge moving in an electromagnetic field. Show that the rate at which Lorentz force does work on a moving charge having velocity v is Fₑ ⋅ v, where Fₑ is electric force.

The Lorentz force describes the force experienced by a charged particle moving through an electromagnetic field. It can be expressed mathematically as:

Lorentz Force Equation

The equation for the Lorentz force \( \mathbf{F} \) acting on a charge \( q \) moving with velocity \( \mathbf{v} \) in an electric field \( \mathbf{E} \) and a magnetic field \( \mathbf{B} \) is given by:

\( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \)

Understanding the Components

• \( \mathbf{E} \): The electric field vector.
• \( \mathbf{B} \): The magnetic field vector.
• \( \mathbf{v} \): The velocity of the charge.
• \( q \): The charge of the particle.

Work Done by the Lorentz Force

To show that the rate at which the Lorentz force does work on a moving charge is given by \( \mathbf{F}_e \cdot \mathbf{v} \), we focus on the electric component of the Lorentz force, \( \mathbf{F}_e = q\mathbf{E} \).

Calculating the Work Rate

The power \( P \) or rate of work done by the electric force on the charge can be expressed as:

\( P = \mathbf{F}_e \cdot \mathbf{v} \)

Substituting \( \mathbf{F}_e \) into this equation gives:

\( P = (q\mathbf{E}) \cdot \mathbf{v} \)

This shows that the work done by the electric force on the charge is indeed \( \mathbf{F}_e \cdot \mathbf{v} \), confirming the relationship between the Lorentz force and the work done on a moving charge.