Question

How much work is required to move a proton and an electron at rest a distance $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{}\mathit{m}$apart to be at rest a distance$\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{}\mathit{m}$ apart?

Short answer

The work required to move the electron and proton is $4.3\times {10}^{-21}\text{J}$.

Step-by-step solution

Write the given data from the question.

Initial distance between electron and proton, $r_1=3\times {10}^{-8}\text{m}$

Final distance between electron and proton, $r_2=7\times {10}^{-8}\text{m}$

Charge on proton, $q_1=+1.6\times {10}^{-19}\text{C}$

Charge of electron,$q_2=-1.6\times {10}^{-19}\text{C}$

Determine the formulas to calculate the work required to move a proton and electron.

The expression to calculate the work required to move proton and electron is given as follows.

$W=9\times {10}^9\times \left|q_1{q}_2\right|\left(\frac{1}{r_2}-\frac{1}{r_1}\right)$ …… (i)

Calculate the work required to move a proton and electron.

Calculate the required work.

Substitute $3\times {10}^{-8}\text{m}$for $r_1$,$1.6\times {10}^{-19}\text{C}$, for $q_1$, $7\times {10}^{-8}\text{m}$for $r_2$, and$-1.6\times {10}^{-19}\text{C}$ for$q_2$ into equation (i).

$$\begin{gathered} W=9\times {10}^9\times \left|1.6\times {10}^{-19}\times \left(-1.6\times {10}^{-19}\right)\right|\left(\frac{1}{3\times {10}^{-8}}-\frac{1}{7\times {10}^{-8}}\right) \\ W=9\times {10}^9\times \frac{2.56\times {10}^{-38}}{{10}^{-8}}\left(\frac{7-3}{3\times 7}\right) \\ W=9\times 2.56\times {10}^{-21}\times \frac{4}{21} \\ W=4.3\times {10}^{-21}\text{J} \end{gathered}$$

Hence the work required to move the electron and proton is $4.3\times {10}^{-21}\text{J}$.