Q28P - Fundamentals Of Physics

Question

$$\begin{gathered} \mathbf{A}\mathbf{ }\mathbf{particle}\mathbf{ }\mathbf{undergoes}\mathbf{ }\mathbf{uniform}\mathbf{ }\mathbf{circular}\mathbf{ }\mathbf{motion}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{radius}\mathbf{ }\mathbf{26}\mathbf{.}\mathbf{1}\mathbf{ }\mathbf{\mu }\mathbf{m}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{uniform}\mathbf{ }\mathbf{magnetic}\mathbf{ }\mathbf{field}\mathbf{.}\mathbf{ } \\ \mathbf{The}\mathbf{ }\mathbf{magnetic}\mathbf{ }\mathbf{force}\mathbf{ }\mathbf{on}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{particle}\mathbf{ }\mathbf{has}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{magnitude}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{60}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{17}}\mathbf{ }\mathbf{ }\mathbf{N}\mathbf{.} \\ \mathbf{ }\mathbf{What}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{kinetic}\mathbf{ }\mathbf{energy}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{particle}\mathbf{?} \end{gathered}$$

Step-by-Step Solution

Answer

$\mathrm{The} \mathrm{kinetic} \mathrm{energy} \mathrm{of} \mathrm{the} \mathrm{particle} \mathrm{is} 2.09\times {10}^{-22}\mathrm{J} .$

Step 1: Listing the given quantities

$$\begin{gathered} •\mathrm{Radius} \mathrm{r}=26.1\mathrm{\mu m}=26.1\times {10}^{-6}\mathrm{m}. \\ •\mathrm{Magnetic} \mathrm{force} \mathrm{F}=1.60\times {10}^{-17}\mathrm{N} \end{gathered}$$

Step 2: Understanding the concept of kinetic energy and centripetal force

$$\begin{gathered} \mathbf{We}\mathbf{ }\mathbf{use}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{formula}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{centripetal}\mathbf{ }\mathbf{force}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{derive}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{formula}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{kinetic}\mathbf{ }\mathbf{energy}\mathbf{,} \\ \mathbf{ }\mathbf{and}\mathbf{ }\mathbf{then}\mathbf{,}\mathbf{ }\mathbf{using}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{given}\mathbf{ }\mathbf{values}\mathbf{,} \\ \mathbf{ }\mathbf{we}\mathbf{ }\mathbf{can}\mathbf{ }\mathbf{find}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{kinetic}\mathbf{ }\mathbf{energy}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{particle}\mathbf{.} \\ \mathrm{Formula}: \\ \mathrm{Centripetal} \mathrm{force}, \mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}} \\ \mathrm{Kinetic} \mathrm{energy}, \mathrm{K}=\frac{1}{2}{\mathrm{mv}}^2 \end{gathered}$$

Step 3: Explanation

$$\begin{gathered} Since centripetal force is equal to magnetic force, the expression can be written as: \\ F=\frac{m{v}^2}{r} \\ Fr=m{v}^2 \\ But, we have, \\ KE=\frac{m{v}^2}{2} \\ Therefore, \\ KE=\frac{m{v}^2}{2} \\ =\frac{Fr}{2} \\ =\frac{\left(1.60\times {10}^{-17}\right)\left(26.1\times {10}^{-6}\right)}{2} \\ =2.09\times {10}^{-22}J. \\ Therefore, the kinetic energy of the particle is 2.09\times {10}^{-22} . \end{gathered}$$