Question

Approximately what energy would electrons need to have to be useful as probes that could reveal features as small as ${\mathbf{10}}^{\mathbf{-}\mathbf{18}}\mathbf{}\mathbf{m}$,the approximate range of the weak force?

Short answer

The energy would the electrons need to have to be useful as probes that could reveal features as small as${10}^{-18}m$, the approximate range of the weak force is $1.2TeV$.

Step-by-step solution

Given data 

The size of an object, $\lambda ={10}^{-18}m$.

Concept of Energy of electron

The energy of an electron is given as,

$E^2=m^2{c}^4+p^2{c}^2$

$$\begin{gathered} E^2\approx p^2{c}^2 \\ E\approx \frac{h}{\lambda }c \end{gathered}$$

Where is the mass of an electron, $9.1\times {10}^{-31}\text{kg}$,h is reduced Planck's constant, $6.63\times {10}^{-34}\text{J}·\text{s}$, $\lambda$is the size of an object to be examined and is the velocity of light, $3\times {10}^8\text{m/s}$.

Calculation of the energy for an electron 

Substitute the numerical values in equation (1) as:

$\begin{array}{rcl}E& =& \frac{\left(6.63\times {10}^{-34}J·s\right)\left(3\times {10}^{8}m/s\right)}{\left({10}^{-18}m\right)}\\ E& =& 1.989\times {10}^{-7}J\\ & =& 1.2TeV\\ & & \end{array}$

The energy would the electrons need to have to be useful as probes that could reveal features as small as${10}^{-18}m$ , the approximate range of the weak force is $1.2TeV$.