Question

A proton traveling at$\mathbf{23}\mathbf{.}{\mathbf{0}}^{\mathbf{0}}$with respect to the direction of a magnetic field of strength$\mathbf{2}\mathbf{.}\mathbf{60}\mathbf{}\mathbf{mT}$experiences a magnetic force of$\mathbf{6}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{17}}$N. (a)Calculate the proton’s speed. (b)Find its kinetic energy in electron-volts.

Short answer

a. Velocity of proton is$4.00\times {10}^5\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.

b. Kinetic energy of the proton is$835ev$

Step-by-step solution

Given

$$\begin{gathered} \varnothing =23.0^0 \\ B=2.60mT=2.60\times {10}^{-3}T \\ {F}_B=6.50\times {10}^{-17N} \end{gathered}$$

Determining the concept

Find the velocity of the proton using the formula for magnetic field in terms of charge and velocity of proton. Then using the formula for K.E, find the kinetic energy of the proton.

Formulae are as follow:

$$\begin{gathered} F_B=evB\sin \varnothing \\ K.E.=\frac{1}{2}m{v}^2 \end{gathered}$$

Where, F_B is magnetic force, v is velocity, m is mass, K.E. is kinetic energy, B is magnetic field, e is charge of particle.

(a) Determining the proton’s speed 

The magnetic force experienced by the proton is,

$F_B=evB\sin \varnothing$

Hence, velocity of proton is,

$$\begin{gathered} v=\frac{F_B}{eB\sin \varnothing } \\ v=\frac{6.50\times {10}^{-17}}{\left(1.6\times {10}^{-19}\right)\left(2.60\times {10}^{-3}\right)\sin \left(23.0^0\right)} \\ v=399891~4.00\times {10}^5\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

Hence, the velocity of proton is $4.00\times {10}^5\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.

(b) Determining the kinetic energy of the proton

Kinetic energy of the proton is,

$$\begin{gathered} K.E=\frac{1}{2}m{v}^2 \\ K.E=\frac{1}{2}\left(1.67\times {10}^{-27}\right){\left(399891\right)}^2 \\ K.E=1.3352\times {10}^{-16}J \\ K.E=\frac{1.3352\times {10}^{-16}}{1.6\times {10}^{-19}} \\ K.E=834.5~835eV \end{gathered}$$

Hence, the kinetic energy of the proton is 835 eV .

Therefore, by using the formula for magnetic field in terms of charge and velocity of proton and the formula for K.E, the velocity and kinetic energy of proton can be determined.