The electrostatic force $\left(\vec{F_1}\right)$ and magnetic force $\left(\vec{F}_2\right)$ acting on a charge $q$ moving with velocity $v$ can be written :

<p>The correct expressions for the electrostatic force, $\vec{F}_1$, and the magnetic force, $\vec{F}_2$, acting on a charge $q$ moving with velocity $\vec{v}$, are given by the Lorentz force law. This law states that the total force acting on a charged particle in both an electric field and a magnetic field is the sum of an electrostatic force due to the electric field and a magnetic force due to the magnetic field.</p> <p>The electrostatic force is given by:</p> <p>$\vec{F}_1 = q \vec{E}$</p> <p>where $\vec{E}$ is the electric field.</p> <p>The magnetic force is given by:</p> <p>$\vec{F}_2 = q(\vec{v} \times \vec{B})$</p> <p>where $\vec{B}$ is the magnetic field, and $\times$ denotes the cross product, indicating that the magnetic force is perpendicular both to the direction of the velocity of the charge and the direction of the magnetic field.</p> <p>Therefore, the correct option is:</p> <p>Option C: $\vec{F}_1=q \vec{E}, \vec{F}_2=q(\vec{V} \times \vec{B})$</p> <p>Options A, B, and D are incorrect because they do not accurately reflect the definitions of electrostatic and magnetic forces as described by the Lorentz force law.</p>