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# 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.

4.20 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 $9.0 \times 10 ^{-5} V m ^{-1}$, make a simple guess as to what the beam contains. Why is the answer not unique?

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$\\qE=qvB\\ E=vB$

(i)

Let the beam consist of particles having charge q and mass m.

After being accelerated through a potential difference V its velocity can be found out by using the following relation,

$\\\frac{1}{2}mv^{2}=qV\\ \\v=\sqrt{\frac{2qV}{m}}$(ii)

Using the value of v from equation (ii) in (i) we have

$\\E=B\sqrt{\frac{2qV}{m}}\\ \frac{q}{m}=\frac{E^{2}}{2VB^{2}}\\ \frac{q}{m}=\frac{(9\times 10^{-5})^{2}}{2\times 15\times 10^{3}\times (0.75)^{2}}=4.8\times 10^{-13}$

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