Question

A uniform magnetic field \(B\) exists in the region between \(x=0\) and \(x=\frac{3 R}{2}\) (region 2 in the figure) pointing normally into the plane of the paper. A particle with charge \(+Q\) and momentum \(p\) directed along \(x\) -axis enters region 2 from region 1 at point \(P_{1}(y=-R)\). Which of the following option(s) is/are correct? [A] For \(B>\frac{2}{3} \frac{p}{Q R}\), the particle will re-enter region 1 [B] For \(B=\frac{8}{13} \frac{p}{Q R}\), the particle will enter region 3 through the point \(P_{2}\) on \(x\) -axis [C] When the particle re-enters region 1 through the longest possible path in region 2, the magnitude of the change in its linear momentum between point \(P_{1}\) and the farthest point from \(y\) -axis is \(p / \sqrt{2}\) [D] For a fixed \(B\), particles of same charge \(Q\) and same velocity \(v\), the distance between the point \(P_{1}\) and the point of re-entry into region 1 is inversely proportional to the mass of the particle

Short answer

[A] is correct, [B] is incorrect, [C] is incorrect because the magnitude of the change in momentum is not \( p / \sqrt{2} \), and [D] is incorrect because the distance of re-entry is directly proportional to the mass of the particle for a fixed \( B \), \( Q \), and \( v \).

Step-by-step solution

Analyze Options Based on Magnetic Force Concepts

First, consider the magnetic force on a charged particle: the Lorentz force law states that the force on the particle is given by the cross product of the charge's velocity and the magnetic field. The magnetic force is perpendicular to both the velocity of the particle and the magnetic field, and its magnitude is given by the equation \( F = QvB \) where \( v \) is the velocity of the particle, \( Q \) is the charge, and \( B \) is the magnitude of the uniform magnetic field. When a charged particle enters a magnetic field, it experiences this force which causes it to follow a circular path if the field is perpendicular to its velocity. The radius of the circular path \( r \) is given by the relationship \( r = \frac{p}{QB} \), where \( p \) is the linear momentum of the particle. Considering this information, let's evaluate the given options.

Evaluate Option [A]

For the particle to re-enter region 1 after entering region 2, it must not complete a semi-circular path, which would take it to region 3. This implies the radius of curvature in region 2 must be less than or equal to \( \frac{3R}{2} \). Substituting \( r = \frac{3R}{2} \) into \( r = \frac{p}{QB} \) gives \( B \geq \frac{2}{3} \frac{p}{QR} \). Therefore, option [A] is correct.

Evaluate Option [B]

If the particle enters region 3 through point \( P_2 \) on the x-axis, this implies that the particle has completed a semi-circular path. Therefore, the radius of curvature in region 2 must be exactly \( \frac{3R}{2} \), and substituting this into \( r = \frac{p}{QB} \) yields \( B = \frac{2}{3} \frac{p}{QR} \), not \( \frac{8}{13} \frac{p}{QR} \) as stated in option [B]. Thus, option [B] is incorrect.

Evaluate Option [C]

The change in momentum \( \Delta p \) is given by the difference between the initial and final momenta in magnitude and direction. Since the motion is circular and the particle is charged, its direction changes but not its speed, meaning the change in momentum must be perpendicular to the initial momentum. Under these conditions, the longest possible path forms a right-angled triangle where the final and initial momentum vectors form the legs, and the hypotenuse is the diameter of the circular path. Using the Pythagorean theorem, \( \Delta p = \sqrt{p^2 - (p/\sqrt{2})^2} = p/\sqrt{2} \). Therefore, option [C] is incorrect because the correct magnitude of the change in momentum is \( p(1 - 1/\sqrt{2}) \).

Evaluate Option [D]

The distance between point \( P_1 \) and the re-entry point into region 1 depends on the radius \( r \) of the circular path, which is given by \( r = \frac{p}{QB} \). If the charge \( Q \) and the magnetic field \( B \) are fixed, the radius of the path, and therefore the distance of re-entry, depends only on the momentum \( p \), which is the product of mass \( m \) and velocity \( v \). Since \( p = mv \) and \( v \) is constant for the particles in the problem statement, the distance of re-entry into region 1 is directly proportional to the mass \( m \), not inversely proportional. Thus, option [D] is incorrect.

Key concepts

Lorentz Force Law

The Lorentz force law is a cornerstone in understanding how charged particles behave in electromagnetic fields. This principle states that a charged particle moving through a magnetic field experiences a force that is perpendicular to both the velocity of the particle and the direction of the magnetic field. The magnitude of this force can be calculated using the formula:
\[ F = QvB \]
where \( F \) is the magnetic force, \( Q \) is the charge of the particle, \( v \) is the velocity of the particle, and \( B \) is the magnetic field strength.

The direction of the force is given by the right-hand rule, which tells us that if you point your thumb in the direction of the velocity of a positive charge and your fingers in the direction of the magnetic field, your palm points in the direction of the force. This law explains the initial motion of a charged particle as it moves into a uniform magnetic field, and it's the basis for analyzing motion, like in the given exercise, where the charged particle moves from one region into another where a magnetic field is present.

Circular Motion in Magnetic Field

When a charged particle moves into a magnetic field perpendicularly, it starts to follow a circular motion. This happens because the Lorentz force acts as the centripetal force required for circular motion. The radius of this circular motion is linked to the physical properties of the particle and the strength of the magnetic field. Specifically, the radius \( r \) is determined by the formula:
\[ r = \frac{p}{QB} \]
where \( p \) is the momentum of the particle, and the other symbols have their meanings as discussed earlier. In the context of the exercise, understanding circular motion in a magnetic field helps us predict the path of the charged particle within the magnetic field region and to determine whether it will re-enter the first region or move to another, as this depends on the curvature of the particle's path in relation to the size of the magnetic field region.

Momentum Change in Magnetic Field

The concept of momentum change in a magnetic field is essential when considering a charged particle's motion path. Although the speed of the particle remains unchanged since the magnetic force is always perpendicular to its motion, the direction of the momentum changes. As a result, the momentum vector experiences a change, which can be seen as the particle curves through the magnetic field.

For a particle that moves in a circular path due to a magnetic field, the momentum change (∆p) can be visualized geometrically. Considering it follows a semi-circular path before re-entering a region, the initial and final momentum vectors point in opposite directions. This results in a momentum change that equals twice the particle's momentum since it is a 180-degree change in direction. In a scenario like the one in our exercise where the path is not a complete semi-circle, one would use vector subtraction to determine the change. This understanding aids in answering questions about the particle's behavior after traveling through different magnetic field regions.