A particle of mass \(\mathrm{m}\) and charge q moves with a constant velocity v along the positive \(\mathrm{x}\) -direction. It enters a region containing a uniform magnetic field B directed along the negative \(z\) -direction, extending from \(x=a\) to \(x=b\). The minimum value of required so that the particle can just enter the region \(\mathrm{x}>\mathrm{b}\) is (a) \([(\mathrm{q} \mathrm{bB}) / \mathrm{m}]\) (b) \(q(b-a)(B / m)\) (c) \([(\mathrm{qaB}) / \mathrm{m}]\) (d) \(q(b+a)(B / 2 m)\)

The magnetic force on a charged particle moving with a velocity in a magnetic field can be given by: \(F = q(\mathbf{v} \times \mathbf{B})\)

As the magnetic force is the only force acting on the charged particle in the region, the total acceleration (a) can be found using Newton's Second Law: \(F = m \times a\)

Now, we need to find the time taken by the particle to cross the region with a magnetic field (from x = a to x = b) and just enter x > b. We will use the equation of motion, which is: \(x_f = x_0 + v_0t + \frac{1}{2}at^2\)

Using the force equation and equation of motion, we can solve for the minimum velocity required to just enter the region x > b.

Lastly, we will compare our derived expression for the minimum velocity with the given expressions (a), (b), (c), and (d) and choose the appropriate one. Now, let's perform the calculations: 1. The magnetic force acting on the charged particle is given by: \(\mathbf{F} = \mathrm{q}(\mathrm{v} \times \mathrm{B})\), and it is perpendicular to both velocity and magnetic field. The force magnitude becomes: \(F = \mathrm{q}vb\) 2. Calculate the acceleration of the particle due to magnetic force: \(a = F / m = \mathrm{q}vb / \mathrm{m}\) 3. We are given that the particle is traveling along the x-axis and entering a magnetic field region between x = a and x = b. So, the initial position \(x_0 = a\), and the final position \(x_f = b\). Applying the equation of motion for the x-axis, \(b = a + vt + \frac{1}{2}(\frac{qvb}{m})t^2\) 4. Rearranging the equation and solving for v, we get: \(v = \frac{m}{qb}(b - a)\) 5. Comparing with the given expressions, we can see that our derived expression for the minimum velocity matches expression (b): \(v = q(b - a)(B / m)\) So, the correct answer is (b) \(q(b - a)(B / m)\).