Question

At a particular instant a proton exerts an electric force of $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{5}\mathbf{.}\mathbf{76}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{13}}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\hspace{0.33em}}\mathbf{N}$on an electron. How far apart are the proton and the electron?

Short answer

The proton and the electron are$2.001\times {10}^{-8}\hspace{0.33em}m$ apart.

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The proton exerts an electric force of$(0,5.76\times {10}^{-13},0)\hspace{0.33em}N$

Significance of the Coulomb’s law for the distance

The coulomb’s law states that the force exerted by an object is inversely proportional to their distance’s square and directly proportional to the product of their charges.

The equation of the force gives the distance amongst the proton and the electron.

Determination of the distance between proton and electron

From Coulomb’s law, the electric force on the electron by the proton is expressed as:

$F=k\frac{q_1{q}_2}{r^2}$

Here, F is the magnitude of the electric force that is $5.76\times {10}^{-13}\hspace{0.33em}\mathrm{N}$; k is the coulomb’s constant that is about $8.99\times {10}^9\mathrm{N}·{\mathrm{m}}^2/{\mathrm{C}}^2,q_1\text{and}{q}_2$are the charges of the proton and the electron that are $+1.602\times {10}^{-19}\mathrm{C}\text{and}-1.602\times {10}^{-19}\mathrm{C}$and r is the distance amongst them.

Substituting the values in the above equation, we get-

role="math" localid="1658117645354" $$\begin{gathered} 5.76\times {10}^{-13}\mathrm{N}=8.99\times {10}^9\mathrm{N}·{\mathrm{m}}^2/{\mathrm{C}}^2\times \frac{\left(+1.602\times {10}^{-19}\mathrm{C}\right)\times \left(-1.602\times {10}^{-19}\mathrm{C}\right)}{(r{)}^2} \\ {r}^2=\frac{\left(8.99\times {10}^9{\mathrm{N}}^2{\mathrm{m}}^2/{\mathrm{C}}^2\right)\times \left(+1.602\times {10}^{-19}\mathrm{C}\right)\times \left(-1.602\times {10}^{-19}\mathrm{C}\right)}{\left(5.76\times {10}^{-13}\mathrm{N}\right)} \\ {r}^2=4.005\times {10}^{-16}\hspace{0.33em}{m}^2 \\ r=2.001\times {10}^{(}-8)\hspace{0.33em}m \end{gathered}$$

Thus, the proton and the electron are$2.001\times {10}^{-8}m$apart.