Problem 1939 - JEE Physics

Question

An electron moving with a speed \(y_{0}\) along the positive \(x\) -axis at \(\mathrm{y}=0\) enters a region of uniform magnetic field \(\mathrm{B}^{-}=-\mathrm{B}_{0} \mathrm{k} \wedge\) which exists to the right of y-axis. The electron exits from the region after some time with the speed at co-ordinate y then.

Step-by-Step Solution

Answer

The electron, initially moving along the positive x-axis, enters a region with a uniform magnetic field, leading to a Lorentz force causing it to move in a circular path in the xy-plane. The radius of this circular path is given by \(r = \frac{m_ev}{|q|B}\), where \(m_e\) is the mass of the electron, v is its speed, q is its charge, and B is the magnetic field. The electron covers half of the circular path and exits the region with an angle of rotation θ = π radians. Using the angle θ and the radius r, we can find the coordinate y where the electron exits the region as \(y = r \sin(θ)\). Thus, the electron exits at coordinate \(y = 0\).

Step 1: Determine the Lorentz force acting on the electron

The Lorentz force F acting on an electron moving with a velocity v in a magnetic field B is given by: \[F = q(v \times B)\] where q is the charge of the electron. Since the electron is moving along the x-axis and the magnetic field is in the -k direction, the magnetic force will result in a centripetal force causing the electron to move in a circular path in the xy-plane.

Step 2: Calculate the centripetal force and radius

The centripetal force F_c acting on an electron moving in a circular path of radius r is given by: \[F_c = \frac{m_ev^2}{r}\] where me is the mass of the electron and v is the speed. Since the Lorentz force F is acting as the centripetal force, we can equate the two forces and solve for the radius r: \[F_c = F \Rightarrow \frac{m_ev^2}{r} = q(v \times B)\] From this, we get the radius r: \[r = \frac{m_ev}{|q|B}\]

Step 3: Find the angle of rotation θ and the coordinate y

Since the electron exits the region with the magnetic field, it means that it completes (or covers) half of the circular path. Assuming the electron enters the region at t=0, the angle θ covered by the electron in the given time is given by: \[θ = \frac{vt}{r}\] Since the electron covers half of the circular path, its angle of rotation θ is π radians. Now, we can use the angle θ and the radius r to find the coordinate y where the electron exits the region: \[y = r \sin(θ)\]

Step 4: Calculate the final coordinate y

Substituting the angle θ and radius r into the equation for y, we get: \[y = \frac{m_ev}{|q|B} \sin(\pi)\] Since sin(π) = 0, we get: \[y = 0\] So, the electron exits from the region with the magnetic field at coordinate y = 0.

Key Concepts

Lorentz Force

When an electron moves through a magnetic field, it experiences a force known as the Lorentz force. The equation for the Lorentz force is: \[ F = q(v \times B) \] where

• \( q \) is the charge of the electron,
• \( v \) is its velocity, and
• \( B \) is the magnetic field strength.

This force is perpendicular to both the direction of the velocity and the magnetic field. Thus, it changes the electron's direction without altering its speed. Because of this, it doesn't do any work on the electron. This is a key characteristic of motion in a magnetic field, leading to unique effects such as circular motion. To fully understand how this works, it's important to see how the force acts as a centripetal force.

Centripetal Force

The motion of an electron in a magnetic field becomes circular because of the centripetal force. This force is necessary to keep an object moving in a circle, and it's directed towards the center of the path. The centripetal force acting on an electron is given by:\[ F_c = \frac{m_e v^2}{r} \]where

• \( m_e \) is the mass of the electron,
• \( v \) is its velocity, and
• \( r \) is the radius of the path.

In this scenario, the Lorentz force, a magnetic force, acts as the centripetal force. This relationship changes the direction but not the speed of the electron. Understanding how these forces interact helps explain the path of the electron within the magnetic field.

Circular Motion

When an electron moves through a magnetic field, it follows a curved path known as circular motion. This happens because the Lorentz force is always perpendicular to the electron's velocity, causing the path to bend into a circle. The speed remains constant because the force does not work on the electron. The circular motion questioned in physics involves examining the factors contributing to the circular path, which include:

• The magnetic field strength, which influences the force,
• The charge and velocity of the electron,
• And the surrounding mass, which determines how easily the path curves.

By grasping these elements, we can predict how an electron will behave under different flux densities and directions.

Magnetic Field Effect

The effect of a magnetic field on an electron is to influence its motion generally, causing it to curve rather than travel in a straight line. The magnetic field exerts a Lorentz force on the moving electron, fundamentally altering its trajectory. This force depends on several factors:

• The velocity of the electron can change the direction of the force.
• The strength and orientation of the magnetic field directly affect the magnitude and direction of the resulting force.
• The nature of the particle, here being an electron, determines the charge involved in calculating the force.

These elements collectively result in an angular deviation, affecting the electron's path and eventually its exit point from the magnetic field region.

Radius of Path

The radius of the circular path taken by the electron in a magnetic field is significant because it indicates how sharply the electron's path curves. The relationship between the radius and other factors is given by:\[ r = \frac{m_e v}{|q|B} \]where

• \( m_e \) is the electron's mass,
• \( v \) is its velocity,
• \( |q| \) is the magnitude of its charge, and
• \( B \) is the magnetic field strength.

This formula shows the inverse relationship between the radius and the strength of the magnetic field. With a stronger magnetic field, the path curves more sharply, resulting in a smaller radius. Understanding this connection helps in examining how various conditions affect the trajectory of electrons when subjected to magnetic fields.