Question

A charged particle having mass \({\rm{6}}{\rm{.64 \times 1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{\;kg}}\) (that of a helium atom) moving at \({\rm{8}}{\rm{.70 \times 1}}{{\rm{0}}^{\rm{5}}}{\rm{\;m/s}}\) perpendicular to a \({\rm{1}}{\rm{.50 - T}}\) magnetic field travels in a circular path of radius \({\rm{16}}{\rm{.0\;mm}}\).(a)What is the charge of the particle?(b)What is unreasonable about this result?(c)Which assumptions are responsible?

Short answer

1. The charge of the particle is \({\rm{2}}{\rm{.41}} \times {\rm{1}}{{\rm{0}}^{{\rm{ - 19}}}}\;{\rm{C}}\).
2. Theunreasonableobjects must have charges that are integer multiples of the electrical charge.
3. The radius specified does not match the other specifications.

Step-by-step solution

Concept Introduction

The area of space near a magnetic body or a current-carrying body where magnetic forces caused by the body or current can be detected.

Find the value of the charged particle

(a)

The equation\({\rm{r = }}\frac{{{\rm{mv}}}}{{{\rm{qB}}}}\)gives the radius of the curved route in terms of the charge; Apply this equation with the supplied numbers to solve for\({\rm{q}}\).

\(\begin{aligned}{}{\rm{q}} &= \frac{{{\rm{mv}}}}{{{\rm{rB}}}}\\ &= \frac{{\left( {{\rm{6}}{\rm{.64 \times 1}}{{\rm{0}}^{{\rm{ - 27}}}}{\rm{\;kg}}} \right){\rm{ \times }}\left( {{\rm{8}}{\rm{.70 \times 1}}{{\rm{0}}^{\rm{5}}}\;{\rm{m/s}}} \right)}}{{\left( {{\rm{16}}{\rm{.0 \times 1}}{{\rm{0}}^{{\rm{ - 3}}}}\;{\rm{m}}} \right){\rm{ \times }}\left( {{\rm{1}}{\rm{.50\;T}}} \right)}}\\ &= {\rm{2}}{\rm{.41 \times 1}}{{\rm{0}}^{{\rm{ - 19}}}}\;{\rm{C}}\end{aligned}\)

Therefore,The charge of the particle is \({\rm{2}}{\rm{.41 \times 1}}{{\rm{0}}^{{\rm{ - 19}}}}\;{\rm{C}}\).

Explaining the unreasonable result

(b)

When you divide the charge in component\({\rm{2}}{\rm{.41 \times 1}}{{\rm{0}}^{{\rm{ - 19}}}}\;{\rm{C}}\)by the electronic charge, you get

\(\frac{{{\rm{2}}{\rm{.41 \times 1}}{{\rm{0}}^{{\rm{ - 19}}}}\;{\rm{C}}}}{{{\rm{1}}{\rm{.60 \times 1}}{{\rm{0}}^{{\rm{ - 19}}}}\;{\rm{C}}}} = \frac{{\rm{3}}}{{\rm{2}}}\)

Objects must have charges that are integer multiples of the electrical charge.

Explaining which assumptions are responsible

(c)

The radius in question does not fit with the other parameters to provide a suitable charge value.