Question

The electrostatic force \( \mathbf{F_1} \) and magnetic force \( \mathbf{F_2} \) acting on a charge \( q \) moving with velocity \( v \) can be written as:

• A. \( \vec{F_1} = q \vec{v} \cdot \vec{E}, \vec{F_2} = q(\vec{B} \cdot \vec{v}) \)
• B. \( \vec{F_1} = q \vec{B}, \vec{F_2} = q(\vec{B} \times \vec{v}) \)
• C. \( \vec{F_1} = q \vec{E}, \vec{F_2} = q(\vec{v} \times \vec{B}) \)
• D. \( \vec{F_1} = q \vec{E}, \vec{F_2} = q(\vec{B} \times \vec{v}) \)

Hint:

The Lorentz force is the combination of electrostatic and magnetic forces acting on a charge.

Answer 1

The Correct Option is C

{Understanding the Forces} 
The electrostatic force on a charge \( q \) is given by: \[ \vec{F_1} = q \vec{E} \] The magnetic force on a moving charge is given by: \[ \vec{F_2} = q (\vec{v} \times \vec{B}) \] Thus, the correct answer is (C). 
 

Answer 2

Step 1: Understand the context of electromagnetic force on a charge
A charged particle moving in an electric and magnetic field experiences forces due to both fields.
These forces are fundamental interactions described by classical electromagnetism.

Step 2: Electrostatic force
The electric field \( \vec{E} \) exerts a force on a charge \( q \) given by:
\( \vec{F_1} = q \vec{E} \)
This is called the electrostatic force.
It acts in the direction of the electric field for a positive charge, and opposite for a negative charge.

Step 3: Magnetic force
When a charge \( q \) moves with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \), it experiences a magnetic force given by the vector cross product:
\( \vec{F_2} = q(\vec{v} \times \vec{B}) \)
This force is perpendicular to both the velocity vector and the magnetic field, as per the right-hand rule.
It depends on the magnitude of \( v \), the strength of \( B \), and the sine of the angle between them.

Step 4: Combine both forces (Lorentz Force Law)
The total electromagnetic force \( \vec{F} \) acting on a moving charge in electric and magnetic fields is given by:
\( \vec{F} = \vec{F_1} + \vec{F_2} = q\vec{E} + q(\vec{v} \times \vec{B}) \)

Final Answer:
\( \vec{F_1} = q \vec{E}, \vec{F_2} = q(\vec{v} \times \vec{B}) \)