Question

(a)Find the frequency of revolution of an electron with an energy of 100 eV in a uniform magnetic field of magnitude $\mathbf{}\mathbf{35}\mathbf{.0}\mathbf{}\mathit{\mu }\mathit{T}$.

(b)Calculate the radius of the path of this electron if its velocity is perpendicular to the magnetic field.

Short answer

a) Frequency of revolution of an electron is$9.79\times {10}^{5}Hz$.

b) Radius of the path of the electron is $0.9641m$.

Step-by-step solution

Listing the given quantities 

• Energy is .$100eV=100\times 1.60\times {10}^{-19}J$
• The magnitude of the magnetic field is $35.0\mu T$.

Understanding the concept of centrifugal force and energy  

Here we have to usetheconcept of equilibrium of magnetic force and centrifugal force. From this, we can find radius as well as frequency. To find the radius, first, we need to find velocity usingtheequation of kinetic energy.

Formula:

$F_B=qvB$

$F_c=\frac{m{v}^2}{r}$

$KE=0.5m{v}^2$

(a) Calculation of the frequency of revolution of an electron

We havethe magnetic and centrifugal forces, and in equilibrium, both balance each other. That can be written as:

$\begin{array}{c}qvB=\frac{m{v}^2}{r}\\ qB=\frac{mv}{r}\\ \frac{qB}{m}=\frac{v}{r}\\ \frac{qB}{m}=\omega \\ 2\pi f=\frac{qB}{m}\end{array}$

Substitute the values, and we get,

$\begin{array}{c}f=\frac{qB}{2\pi m}\\ =\frac{1.6\times {10}^{-10}\times 35\times {10}^{-6}}{2\times \pi \times 9.1\times {10}^{-31}}\\ =9.79\times {10}^{5}Hz\end{array}$

Thus, the frequency of revolution of an electron is $9.79\times {10}^{5}Hz$.

(b) Calculation of the radius of the path of an electron 

Now we can find thevelocity of an electron using the equation of kinetic energy as follows:

$\begin{array}{c}KE=0.5\times m{v}^2\\ 100\times 1.60\times {10}^{-19}=0.5\times 9.1\times {10}^{-31}\times v^2\\ v=5.93\times {10}^6\text{\hspace{0.17em}m/s}\end{array}$

And

$qvB=\frac{m{v}^2}{r}$

$qB=\frac{mv}{r}$

Substitute the values, and we get,

$\begin{array}{c}r=\frac{mv}{qB}\\ =\frac{9.1\times {10}^{-31}\times 5.93\times {10}^6}{1.6\times {10}^{-19}\times 35\times {10}^{-6}}\\ =0.9641m\end{array}$

Thus, the radius of the path of the electron is$0.9641m$.