91Ó°ÊÓ

Charged particles are fundamental to understanding various physical phenomena, particularly in electromagnetism. A charged particle, such as a proton or electron, has an electric charge that interacts with electric and magnetic fields.

When a charged particle moves through a magnetic field, it experiences a magnetic force. This force can change the direction of the particle's velocity if the force is not zero. The magnetic force's magnitude is given by the formula:

• \[ F = qvB \sin \theta \], where
  • \( F \) is the magnetic force.
  • \( q \) is the charge of the particle.
  • \( v \) is the velocity of the particle.
  • \( B \) is the magnetic field induction.
  • \( \theta \) is the angle between the velocity vector and the magnetic field.

For a particle to pass straight through a magnetic field without deflection, \( \sin \theta \) must be zero (meaning \( \theta = 0^{\circ} \) or \( 180^{\circ} \)), or the product \( qvB \) must equal zero. In many problems, \( \theta \) is typically \( 90^{\circ} \), implying a perpendicular interaction with the magnetic field.

Gravitational force is a universal force that acts between any two masses. In the context of this exercise, it is essential to understand how gravitational force balances other forces like magnetic force to maintain equilibrium. The gravitational force on an object is given by:

• \( F_g = mg \), where
  • \( F_g \) is the gravitational force.
  • \( m \) is the mass of the object.
  • \( g \) is the acceleration due to gravity.

In this scenario, ensuring the particle travels straight without deflection implies that the gravitational force is perfectly balanced by the magnetic force. This condition can be represented by setting \( qvB = mg \). Thus, understanding both forces at play and how they counterbalance is crucial to solving the problem.

While magnetic force tries to alter the motion of the particle, gravitational force constantly acts downward. To maintain a no-deflection path, these forces must harmonize, influencing the velocity necessary to keep the particle on a straight path through the field.

Particles moving through magnetic fields often require specific velocities to achieve desired paths. For example, calculating this velocity is critical when ensuring that a charged particle moves through a magnetic field without deflection.

To find the necessary velocity in our problem, we use the balance between magnetic and gravitational forces given by the equation: \[ qvB = mg \]Rearranging for velocity, we find: \[ v = \frac{mg}{qB} \]
Substituting in the values from our exercise: mass \( m = 10^{-3} \) kg, gravitational acceleration \( g = 10 \text{ m/s}^2 \), charge \( q = 10^{-3} \) C, and magnetic field \( B = 1 \) T, the velocity \( v \) is calculated as follows:

• \[ v = \frac{10^{-3} \times 10}{10^{-3} \times 1} = \frac{10^{-2}}{10^{-3}} = 10 \text{ m/s} \]

This velocity ensures the particle moves without deflection through the magnetic field. Understanding these calculations not only helps in solving textbook exercises but also in practical applications where balancing forces is essential.