Question

How much work must be done to increase the speed of an electron from rest to (a) 0.500c, (b) 0.990c, and (c) 0.9990c?

Short answer

(a) The work for increasing the speed of rest electron is $1.27\times {10}^{-14}\mathrm{J}$.

(b) The work for increasing the speed of rest electron is $5\times {10}^{-13}\mathrm{J}$.

(c) The work for increasing the speed of rest electron is $1.75\times {10}^{-12}\mathrm{J}$.

Step-by-step solution

Identification of given data

The increased speed of electron is ${\mathrm{v}}_{\mathrm{a}}=0.500\mathrm{c}$

The increased speed of electron is ${\mathrm{v}}_{\mathrm{b}}=0.990\mathrm{c}$

The increased speed of electron is ${\mathrm{v}}_{\mathrm{c}}=0.9990\mathrm{c}$

The work necessary to accelerate any particle from rest is equal to the change in kinetic energy of particle from rest to moving position..

Determination of work to accelerate the electron from rest

(a)

The work for increasing the speed of rest electron is given as:

${\mathrm{W}}_{\mathrm{a}}={\mathrm{mc}}^2\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{{\mathrm{v}}_{\mathrm{a}}}{\mathrm{c}}}\right)}^2}}-1\right)$

Here, c is the speed of light and its value is $3\times {10}^8\mathrm{m}/\mathrm{s}$, m is the mass of an electron and its value is $9.1\times {10}^{-31}\mathrm{kg}$.

Substitute all the values in the above equation.

$$\begin{gathered} {\mathrm{W}}_{\mathrm{a}}=\left(9.1\times {10}^{-31}\mathrm{kg}\right){\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{0.500\mathrm{c}}{\mathrm{c}}}\right)}^2}}-1\right) \\ {\mathrm{W}}_{\mathrm{a}}=1.27\times {10}^{-14}\mathrm{J} \end{gathered}$$

Therefore, the work for increasing the speed of rest electron is $1.27\times {10}^{-14}\mathrm{J}$.

Determination of work to accelerate the electron from rest

(b)

The work for increasing the speed of rest electron is given as:

${\mathrm{W}}_{\mathrm{b}}={\mathrm{mc}}^2\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{{\mathrm{v}}_{\mathrm{b}}}{\mathrm{c}}}\right)}^2}}-1\right)$

Substitute all the values in the above equation.

$$\begin{gathered} {\mathrm{W}}_{\mathrm{b}}=\left(9.1\times {10}^{-31}\mathrm{kg}\right){\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{0.990\mathrm{c}}{\mathrm{c}}}\right)}^2}}-1\right) \\ {\mathrm{W}}_{\mathrm{b}}=5\times {10}^{-13}\mathrm{J} \end{gathered}$$

Therefore, the work for increasing the speed of rest electron is $5\times {10}^{-13}\mathrm{kg}$.

Determination of work to accelerate the electron from rest

(c)

The work for increasing the speed of rest electron is given as:

${\mathrm{W}}_{\mathrm{c}}={\mathrm{mc}}^2\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{{\mathrm{v}}_{\mathrm{c}}}{\mathrm{c}}}\right)}^2}}-1\right)$

Substitute all the values in the above equation.

$$\begin{gathered} {\mathrm{W}}_{\mathrm{c}}=\left(9.1\times {10}^{-31}\mathrm{kg}\right){\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2\left(\frac{1}{\sqrt{1-{\left({\displaystyle \frac{0.9990\mathrm{c}}{\mathrm{c}}}\right)}^2}}-1\right) \\ {\mathrm{W}}_{\mathrm{c}}=1.75\times {10}^{-12}\mathrm{J} \end{gathered}$$

Therefore, the work for increasing the speed of rest electron is $1.75\times {10}^{-12}\mathrm{J}$.