91Ó°ÊÓ

An electric field is a region where charged particles experience a force. In a velocity selector, which is part of the magnetic field setup in the exercise, the electric field plays a crucial role by interacting with charged particles. The selector uses perpendicular electric and magnetic fields to control the path of ions passing through it.
The formula for the electric field in the velocity selector is given by:

• \[ E = vB \]

This relates the electric field \( E \), speed of the ions \( v \), and the magnetic field \( B \). By setting the magnetic and electric forces equal, ions with the correct velocity (4.50 km/s here) pass through without deflection. In this exercise, the electric field was calculated as 112.5 N/C—meaning each coulomb of charge feels a 112.5-newton force. The field direction is essential as it determines which ions proceed directly.
Understanding how the electric field influences ion motion is key to grasping the velocity selector’s functionality.

A velocity selector ensures only ions with a specific speed can pass undeviated by balancing the forces acting on them. It combines an electric field and a magnetic field, positioned perpendicular to one another, to achieve this selectivity. Charged particles entering the selector experience electric \( F_e = qE \) and magnetic \( F_m = qvB \) forces.

• The forces are equal (\[ qE = qvB \])
• This means ions with velocity \[ v \] do not deflect:

If velocities differ, the forces won't cancel, causing the ions to move out of the desired path. This controlled movement through a velocity selector forms the basis of many applications, like mass spectrometers, where precise ion velocity is crucial.

In moving charged particles through magnetic fields, the concept of centripetal force arises when particles curve due to the force acting towards the center of their circular path. For ions in a magnetic field with velocity \( v \), the centripetal force required is provided by the magnetic field. The equation relating these is:

• The magnetic force: \[ qvB = \frac{mv^2}{r} \]

Here, \( m \) is mass, \( v \) is velocity, \( r \) is the radius of curvature, and \( B' \) is magnetic field strength. Ion semicircle motion (radius of 0.125 m) in the second magnetic section displays this force.
By rearranging and solving for \( B' \), we find that the required magnetic field is approximately 1.49 T (Tesla). This force directs the ions along a curved path, keeping them in orbit in practical devices such as cyclotrons and particle accelerators.

Singly ionized ions are atoms that have lost or gained one electron, resulting in a net charge of +1e or -1e. In the given exercise, we consider a potassium ion (\(^{40}K^{+}\)). Having one less electron than the neutral atom, its charge is +1 elementary charge (approximately \( 1.6 \times 10^{-19} \text{ C} \)). This net charge allows ions to be manipulated by electric and magnetic fields.

• Loss of an electron modifies an ion’s interactions within fields, leading to acceleration or deflection.
• This is due to the forces described by the equations \[ F = qE \] and \[ F = qvB \].
• In the velocity selector and deflection scenarios, the charge is significant for calculations and influences their precise behavior.

Understanding the concept of ionization allows better insight into controlled ion trajectories in practical scenarios such as isotope separation or ion beam targeting.