Question

Find the value of \( v_0 \) in terms of \( E \), \( d \), and \( I \).

Hint:

For a charged particle moving in a magnetic and electric field, the balance between the forces determines the velocity of the particle.

Answer 1

The magnetic force \( \mathbf{F_B} \) and the electric force \( \mathbf{F_E} \) must balance each other to maintain constant velocity for the particle. Therefore: \[ F_E = F_B \] Substituting the expressions for electric and magnetic forces: \[ qE = qv_0 B \] Using the formula for the magnetic field produced by a current-carrying conductor at a distance \( d \): \[ B = \frac{\mu_0 I}{2 \pi d} \] Thus, equating the forces: \[ E = v_0 \frac{\mu_0 I}{2 \pi d} \] Solving for \( v_0 \): \[ v_0 = \frac{E 2 \pi d}{\mu_0 I} \] Thus, the value of \( v_0 \) is: \[ v_0 = \frac{E 2 \pi d}{\mu_0 I} \]