The uniform magnetic field in the velocity selector is...

The uniform magnetic field in the velocity selector is directed out of the page and has magnitude $B$. (a) What should the magnitude and direction of the electric field in the selector be to allow ions with speed $v$ to pass straight through? (b) Sketch the trajectory inside the velocity selector for ions that enter with speeds slightly less than $v$.

The uniform magnetic field in the velocity selector is directed out of the page and has magnitude $B$.
(a) What should the magnitude and direction of the electric field in the selector be to allow ions with speed $v$ to pass straight through?
(b) Sketch the trajectory inside the velocity selector for ions that enter with speeds slightly less than $v$.

Key Concepts

Lorentz Force

The Lorentz Force is the net force acting on a charged particle due to both electric and magnetic fields, given by the equation F = q(E + v × B). This fundamental concept explains how charged particles move in electromagnetic fields and is crucial in determining their trajectories in devices like velocity selectors.

Electric Field

An electric field exerts a force on a charged particle that is directly proportional to the particle's charge and the strength of the field. In the context of a velocity selector, the electric field is configured to oppose the magnetic force for particles of a specific velocity, ensuring they pass through un-deflected.

Magnetic Field

A magnetic field affects moving charged particles by exerting a force perpendicular to both the particle’s velocity and the field direction. This force causes the particle to curve into a circular or helical path, and its magnitude and orientation are key to controlling the motion of particles in devices that employ magnetic deflection.

Velocity Selector

A velocity selector uses crossed electric and magnetic fields to filter charged particles by their speed. When the electric and magnetic forces balance each other, only particles with the matching velocity experience zero net force and travel in a straight line, making this concept essential in many analytical and experimental applications.

Equilibrium of Forces

The equilibrium of forces occurs when the electric and magnetic forces acting on a charged particle cancel out, leading to zero net force. This condition is critical in the design of velocity selectors, as it allows only particles with the designated velocity to traverse the selector without deflection.

Particle Trajectory

The trajectory of charged particles in crossed electric and magnetic fields is determined by the balance or imbalance of the forces. When particles enter with velocities different from the one required for equilibrium, they experience a net force that curves their path, resulting in trajectories that deviate from a straight line, such as circular arcs or spirals.

Transcript

00:02 In this given problem, positively charged uranium ions are coming out of the accelerating parallel plates.

00:17 These two equally and oppositely charged plates are being used to accelerate the uranium ions, positively charged ions.

00:33 Now these ions, this beam of ions is allowed to pass through a velocity selector.

00:42 Here this horizontal plate are used as a velocity selector through which this beam is passing undeflected.

00:53 In this region the magnetic field is uniform and coming out of the plane of paper.

01:03 Like this.

01:10 So magnetic field in the velocity selector is out of the plane of paper.

01:42 Now in the first part of the problem we have to find direction of electric field so that this beam of positively charged uranium ions remains may remain undeflected.

01:56 So first of all, here this is the direction of motion of the ions and these are the positively charged ions.

02:05 Velocity v and magnetic field is upward, outward, perpendicular to the plane of paper.

02:13 So using expression for magnetic laurence force experienced by these ions which are moving towards right.

02:25 And that expression for the magnetic lawrence force is f bar is equal to u into v cross b.

02:35 So using right -hand screw rule in this cross -product, we get direction of v cross -b, is the cross -product, and hence same will be the direction of force...

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