A magnetic field set up using Helmholtz coils (described in Exercise 4.16) is uniform in a small region and has a magnitude of 0.75 T. In the same region, a uniform electrostatic field is maintained in a direction normal to the common axis of the coils. A narrow beam of (single species) charged particles all accelerated through 15 kV enters this region in a direction perpendicular to both the axis of the coils and the electrostatic field. If the beam remains undeflected when the electrostatic field is 9.0 × 105 V m–1, make a simple guess as to what the beam contains. Why is the answer not unique?

Now here the charged particle is two forces, magnetic Lorentz force due to magnetic field and Electrostatic force and electrostatic force due to electric field, now for particle to go undeflected, the direction of both the forces should be exactly opposite to each other, and magnitude should be equal. direction of force is same as that of Electric field for a positively charged particle and is exactly reversed in case of negatively charged particle and direction of magnetic force is given by Flemings left hand rule.


The situation is depicted in the figue.



Electrostatic force is given by



Where F is the force experienced by particle having charge q and moving in an electric field E Magnitude of magnetic force on particle moving with velocity v in a magnetic field B, velocity makes angle with the Field



here ,since velocity is perpendicular to magnetic field


so


So Equating both we get



so we get


velocity of the particle,


now the particle is accelerated through a potential difference When an charged particle having charge q is accelerated by a potential difference V volts it gain kinetic energy given by


K = Vq, which imparts it with velocity


We know kinetic energy of a Particle with mass m and velocity v is given by



Equating both equations



We get velocity of the particle as



So Equating both the values of velocity v we get



Solving we get



Here we are given,


Electric field E = 9.0 × 105 V m–1


Magnetic Field B = 0.75 T


The Potential difference V = 15 kV = 15000 V


Putting all the values we get



⇒ q/m = 4.8 107 C/Kg


This is the charge to mass ratio of Deuterium ions Dor deuterons, which is a stable isotope of Hydrogen having one proton and one neutron in its nucleus.


So charge on Deuterium ion = e (since 1 charge)


= C


Mass of Deuterium ion,


m = Kg


so charge to mass ratio,


≈ 4.8 × 107 C/Kg


the answer is not unique because only the ratio of charge to mass is determined. There can Other possible species whose charge to mass ratio is same as this value examples are


He, Li, etc.


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