91Ó°ÊÓ

The given data is listed below as:

• The strength of the breakdown field of air is,$E=3×{10}^6\text{N/C}$
• The radius of the metal sphere in the first case is,$R_1=10\text{cm}=10\text{cm}×\left(\frac{\text{1}\text{m}}{\text{1} 00\text{cm}}\right)=0.1\text{m}$ .
• The radius of the metal sphere in the second case is $R_2=1\text{mm}=1\text{mm}×\left(\frac{\text{1}\text{m}}{\text{1} 000\text{mm}}\right)=0.001\text{m}$.

The electric field is referred to as a region that helps an electrically charged particle to exert force on another particle. The magnitude of the electric field is directly proportional to the charge induced and inversely proportional to the square of their distances.

Electric breakdown happens because of the reason that the electric field’s magnitude of an object that is electrically charged is mainly above the limit of the electrical breakdown of a particular insulator that mainly surrounds an object that is charged.

If the distance amongst the charged object and the point inside the electric field is decreased to a high value, then the electric field’s magnitude will exceed the value of the breakdown limit of the electric field. Hence, it is the reason the breakdown occurs near the surface of the sphere.

Thus, electric breakdown occurs as the magnitude of the electric field is above the limit of the electric breakdown for a charged object. If the distance amongst the charged object and the point inside the electric field is decreased to a high value, then the electric field’s magnitude will exceed the value of the breakdown limit of the electric field, so that electric breakdown occurs near the sphere’s surface.

The equation of the magnitude of the electric field for a metal sphere is expressed as:

$$\begin{gathered} E=k\frac{q}{{R_1}^2} \\ q=\frac{E{R}_1^2}{k} \end{gathered}$$

Here, is the magnitude of the electric field that is the strength of breakdown field of air, is the electric constant with the value , is the excess charge and is the radius of the metal sphere in the first case.

Substitute the values in the above equation.

$$\begin{gathered} q=\frac{\left(3×{10}^6\text{N/C}\right)×{\left(0.1\text{m}\right)}^2}{\left(9×{10}^9{\text{Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)} \\ =\frac{\left(3×{10}^6\text{N/C}\right)×0.01{\text{m}}^2}{\left(9×{10}^9{\text{Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)} \\ =\frac{\left(30000\text{N}·{\text{m}}^2\text{/C}\right)}{\left(9×{10}^9{\text{Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)} \\ =3.33×{10}^{-6}\text{C} \end{gathered}$$

Thus, the excess charge that can be put on a metal sphere is for a metal sphere of radius.

The equation of the magnitude of the electric field for a metal sphere is expressed as:

$$\begin{gathered} E=k\frac{q}{{R_2}^2} \\ q=\frac{E{R}_2^2}{k} \end{gathered}$$

Here, is the magnitude of the electric field that is the strength of breakdown field of air, is the electric constant, is the excess charge and is the radius of the metal sphere in the second case.

Substitute the values in the above equation.

$$\begin{gathered} q=\frac{\left(3×{10}^6\text{N/C}\right)×{\left(0.001\text{m}\right)}^2}{\left(9×{10}^9{\text{Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)} \\ =\frac{\left(3×{10}^6\text{N/C}\right)×\left(1×{10}^{-6}{\text{m}}^2\right)}{\left(9×{10}^9{\text{Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)} \\ =\frac{\left(3\text{N}·{\text{m}}^2\text{/C}\right)}{\left(9×{10}^9{\text{Nm}}^{\text{2}}{\text{/C}}^{\text{2}}\right)} \\ =3.33×{10}^{-10}\text{C} \end{gathered}$$

Thus, the excess charge that can be put on a metal sphere is for a metal sphere of radius.