Question

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

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

Answer 1

The Correct Option is C

The electrostatic force acting on a charged particle is given by:

\[ \vec{F}_1 = q\vec{E}, \]

where:
- \( q \) is the charge of the particle,
- \( \vec{E} \) is the electric field.

The magnetic force acting on a charged particle moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by the Lorentz force law:

\[ \vec{F}_2 = q(\vec{v} \times \vec{B}), \]

where:
- \( q \) is the charge of the particle,
- \( \vec{v} \) is the velocity of the particle,
- \( \vec{B} \) is the magnetic field.

Therefore, the correct expressions for the electrostatic and magnetic forces are:

\[ \vec{F}_1 = q\vec{E}, \quad \vec{F}_2 = q(\vec{v} \times \vec{B}). \]

Hence, the correct option is (3).

Answer 2

Step 1: Recall the forces acting on a moving charge.
A charge \( q \) moving with velocity \( \vec{v} \) in an electromagnetic field experiences two types of forces:
(1) Electrostatic force due to the electric field \( \vec{E} \).
(2) Magnetic force due to the magnetic field \( \vec{B} \).

Step 2: Write the expressions for both forces.
The electrostatic force is given by:
\[ \vec{F_1} = q \vec{E} \]
This force acts in the direction of the electric field for a positive charge and opposite to it for a negative charge.

The magnetic force is given by:
\[ \vec{F_2} = q (\vec{v} \times \vec{B}) \]
This force is perpendicular to both the velocity vector \( \vec{v} \) and the magnetic field \( \vec{B} \), following the right-hand rule.

Step 3: Combine the results.
\[ \vec{F_1} = q \vec{E}, \quad \vec{F_2} = q (\vec{v} \times \vec{B}) \]

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