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

Expression for Lorentz Magnetic Force:
The Lorentz force F⃗\vec{F} on a charged particle of charge qq moving with velocity v⃗\vec{v} in a magnetic field B⃗\vec{B} is given by:
F⃗=q(v⃗×B⃗)\vec{F} = q (\vec{v} \times \vec{B})
Where:
• F⃗\vec{F} is the magnetic force on the charged particle.
• qq is the charge of the particle.
• v⃗\vec{v} is the velocity of the particle.
• B⃗\vec{B} is the magnetic field.
This force is always perpendicular to both the velocity v⃗\vec{v} and the magnetic field B⃗\vec{B}, as it is a result of the cross product v⃗×B⃗\vec{v} \times \vec{B}.
Showing that No Work is Done by the Lorentz Force:
To prove that no work is done by the Lorentz magnetic force on the charged particle, we need to calculate the work done by the force. The work done dWdW by a force F⃗\vec{F} over an infinitesimal displacement ds⃗d\vec{s} is given by:
dW=F⃗⋅ds⃗dW = \vec{F} \cdot d\vec{s}
Since ds⃗=v⃗ dtd\vec{s} = \vec{v} \, dt, the work done can be written as:
dW=F⃗⋅v⃗ dtdW = \vec{F} \cdot \vec{v} \, dt
Substituting the expression for the Lorentz force F⃗=q(v⃗×B⃗)\vec{F} = q (\vec{v} \times \vec{B}), we get:
dW=[q(v⃗×B⃗)]⋅v⃗ dtdW = \left[ q (\vec{v} \times \vec{B}) \right] \cdot \vec{v} \, dt
Now, using the vector identity (A⃗×B⃗)⋅C⃗=A⃗⋅(B⃗×C⃗)(\vec{A} \times \vec{B}) \cdot \vec{C} = \vec{A} \cdot (\vec{B} \times \vec{C}), we observe that the cross product of v⃗×B⃗\vec{v} \times \vec{B} is always perpendicular to the velocity v⃗\vec{v}. Therefore, the dot product (v⃗×B⃗)⋅v⃗(\vec{v} \times \vec{B}) \cdot \vec{v} equals zero because the vectors v⃗\vec{v} and v⃗×B⃗\vec{v} \times \vec{B} are perpendicular to each other.
Thus, we find:
dW=0dW = 0
Since the infinitesimal work done dW=0dW = 0, no work is done by the Lorentz magnetic force over any displacement.
Conclusion:
The Lorentz magnetic force does no work on the charged particle because the force is always perpendicular to the velocity of the particle, and hence the dot product F⃗⋅v⃗\vec{F} \cdot \vec{v} is zero. Therefore, the magnetic force cannot change the kinetic energy of the particle.