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:
- \( \vec{F_1} = q \vec{V} \cdot \vec{E}, \quad \vec{F_2} = q (\vec{B} \cdot \vec{V}) \)
- \( \vec{F_1} = q \vec{B}, \quad \vec{F_2} = q (\vec{B} \times \vec{V}) \)
- \( \vec{F_1} = q \vec{E}, \quad \vec{F_2} = q (\vec{V} \times \vec{B}) \)
- \( \vec{F_1} = q \vec{E}, \quad \vec{F_2} = q (\vec{B} \times \vec{V}) \)
The Correct Option is C
Approach Solution - 1
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).
Approach Solution -2
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: \( \vec{F_1} = q \vec{E}, \quad \vec{F_2} = q (\vec{v} \times \vec{B}) \)
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