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

Constant velocity of the particle is maintained when the magnetic force \( \mathbf{F_B} \) equals the electric force \( \mathbf{F_E} \). This equality is expressed as: \[ F_E = F_B \] The equation becomes: \[ qE = qv_0 B \] Given the magnetic field \( B \) generated by a current-carrying conductor at distance \( d \) is \( B = \frac{\mu_0 I}{2 \pi d} \), we equate the forces: \[ E = v_0 \frac{\mu_0 I}{2 \pi d} \] Solving for \( v_0 \) yields: \[ v_0 = \frac{E 2 \pi d}{\mu_0 I} \] Therefore, \( v_0 \) is calculated as: \[ v_0 = \frac{E 2 \pi d}{\mu_0 I} \]