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).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32