Question

A spring of spring constant ' \(K^{\prime}\) is fixed at one end has a small block of mass \(\mathrm{m}\) and charge \(q\) is attached at the other end. The block rests over a smooth horizontal surface. A uniform and constant magnetic field B exists normal to the plane of paper as shown in figure. An electric field \(\vec{E}=E_{0} \hat{i}\left(E_{0}\right.\) is a positive constant) is switched on at \(\mathrm{t}=0 \mathrm{sec} .\) The block moves on horizontal surface without ever lifting off the surface. Then the normal reaction acting on the block is : (1) Maximum at extreme position and minimum at mean position. (2) Maximum at mean position and minimum at extreme position. (3) Is uniform throughout the motion. (4) Is both maximum and minimum at mean position.

Short answer

3. The normal reaction is uniform throughout the motion.

Step-by-step solution

- Analyze forces acting on the block

The block experiences various forces: the spring force, the electric force due to the electric field, the magnetic force due to the magnetic field, and the normal and gravitational forces. The spring force can be given by Hooke's Law as \( F_s = -Kx \).

- Consider the electric force

The electric force on the block is given by \( \bf{F}_E = q \bm{E} \). Since \( \bm{E} = E_0 \bf{\tell} \), the electric force acting on the block is \( \bf{F}_E = qE_0 \tell \).

- Write the equation for the magnetic force

The magnetic force on the block, considering the block's velocity \(v\) in a magnetic field \(B \) is given by \( \bf{F}_B = q(\bf{v} \times \bf{B}) \). Since \( \bm{B} \) is normal to the plane, \( \bf{F}_B \) acts perpendicularly to both \( B \) and velocity \( v \).

- Understand the oscillating motion

The block performs simple harmonic motion under the influence of the spring force, electric force, and magnetic force. The position of the block varies sinusoidally with time, thus changing the magnitude of the spring force periodically.

- Analyze the vertical forces for normal reaction

The vertical forces include gravitational force \( mg \) and the normal force \( N \). Throughout the motion, there are no vertical forces affecting the block's lift-off, thereby making the normal reaction constant throughout the motion.

- Conclude the behavior of the normal reaction

Since the normal force balances the gravitational force with no varying vertical force component (since all acting forces are either lateral or balanced gravitationally), the normal reaction force remains uniform throughout the motion.

Key concepts

Hooke's Law

Hooke's Law describes the restoring force a spring exerts on an object attached to it. According to Hooke's Law, the force exerted by the spring is proportional to the displacement of the object from its equilibrium position. The formula is given by:
\( F_s = -Kx \) where:

• \(F_s\): The spring force
• \(K\): The spring constant, a measure of the stiffness of the spring
• \(x\): The displacement from the equilibrium position

The negative sign indicates that the direction of the spring force is opposite to the direction of displacement. In our exercise, this spring force plays a crucial role in the oscillatory motion of the block.

Lorentz Force

Lorentz Force describes the force experienced by a charged particle moving through electric and magnetic fields. It is given by the equation:

• n\( \bf{F} = q( \bf{E} + \bf{v} \times \bf{B} ) \)

where:

• \( \bf{F}\): The total Lorentz force
• \( q \): The charge of the particle
• \( \bf{E} \): The electric field
• \( \bf{v} \): The particle's velocity
• \( \bf{B} \): The magnetic field

In our scenario, the block is influenced by both an electric field and a magnetic field. The electric force is given by \( \bf{F}_E = q \bf{E} \) and the magnetic force is \( \bf{F}_B = q(\bf{v} \times \bf{B}) \). These forces contribute to the overall motion of the block.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement. The motion follows a sinusoidal pattern over time. In SHM, the position \( x(t ) \) of an object can be described using the function: \[ x(t) = A \text{cos}( \text{\omega} t + \text{\phi} ) \] where:

• \(A\): The amplitude of the motion
• \( \text{\omega} \): The angular frequency
• \( t \): Time
• \( \text{\phi} \): The phase constant

In the problem at hand, the block attached to the spring undergoes simple harmonic motion due to the interplay of the spring force, electric force, and magnetic force. This results in the block oscillating back and forth around its equilibrium position.

Normal Force Analysis

The normal force is the perpendicular force exerted by a surface on an object in contact with it, opposing the weight of the object. In our exercise, the block is on a smooth horizontal surface experiencing various forces in horizontal directions but no vertical forces act to lift it off. Therefore, the normal force \(N\) remains uniform throughout the motion and is equal to the gravitational force acting on the block, given by: \[ N = mg \] where:

• \( N \): The normal force
• \( m \): The mass of the block
• \( g \): The acceleration due to gravity

Since no vertical forces vary, the normal force maintains a consistent value, balancing the gravitational pull. This consistency ensures that the block remains on the surface, experiencing smooth horizontal motion.