Question

derive an expression foe the velocity vc of a positive ions passing undeflected through a region where crossed and uniform electrical field E and magnetic field B are simultaneously present. draw and justify the trajectory of identical positive ions whose velocity has a magnitude less than IvcI

Answer 1

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials (such as iron) and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges (electric currents) such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field varies with location. As such, it is an example of a vector field.
The term 'magnetic field' is used for two distinct but closely related fields denoted by the symbols B and H. In the International System of Units, H is measured in units of amperes per meter and B is measured in teslas, which are equivalent to newtons per meter per ampere. H and B differ in how they account for magnetization. In a vacuum, Band H are the same aside from units; but in a magnetized material, B/{\displaystyle \mu _{0}} and H differ by the magnetization M of the material at that point in the material.

Answer 2

Solution:

E = Ej  ( y- axis)

B = B k ( z- axis)

Force on the positive ion due to electric field present,  

   F (due to E) = $q \times E_j$

Force on the  positive ion due to magnetic field present,

    F (due to B) = $q (v \times c \times B)k$

To pass undeflected the condition to be satisfied,

     F (due to E) = - F (due to B)

i.e,  q  Ej = $- q (vc \times B)k$

This can be only possible if  $vc = \frac{E}{B} \times I$

Justification:

For positive ions with speed v < vc,

Force due to the electric field F' (due to E) = $q \times E$ = F (due to E)

Force due to the magnetic field F' (due to B) = $q \times v \times B$ < F (due to B) since v < $v \times c.$

Since the forces on the positive ion are unbalanced, it will experience acceleration along E.