22.32: A very small object with mass \(8.20 \times 1{0^{ - 9}}\,kg\) and positive charge \(6.50 \times 1{0^{ - 9}}\,C\) is projected directly towards a very large insulating sheet of positive charge that has uniform surface charge density \(5.90 \times 1{0^{ - 8}}\,C/{m^2}\). The object is initially \(0.400\,m\) from the sheet. What initial speed must the object have in order for its closest distance of approach to the sheet to be \(0.100\,m\)?

• The mass of the object is\(8.20 \times 1{0^{ - 9}}\,kg\).
• The magnitude of the charge of the object is\(6.50 \times 1{0^{ - 9}}\,C\).
• The surface charge density of the sheet is \(5.90 \times 1{0^{ - 8}}\,C/{m^2}\).
• Initial distance of the object from the sheet is \(0.400\,m\).

The expression for the acceleration in terms of surface charge density, charge and mass is given by,

\(a = \frac{{q\sigma }}{{2{\varepsilon _o}m}}\)…… (1)

Here \(q\) is the magnitude of the charge, \(\sigma \) is the surface charge density, \(m\) is the mass of the object, \({\varepsilon _o}\) is the permittivity of free space.

Using the kinematics, equation of motion is given by,

\({v^2} = {u^2} - 2as\)

Overall displacement of the object,

\(\begin{aligned}{c}s = 0.4 - 0.1\\ = 0.3\,m\end{aligned}\)

Since the object is trying to approach the large insulating sheet, so the final speed of the object is zero,

\(\begin{aligned}0 = {u^2} - 2\frac{{q\sigma }}{{2{\varepsilon _o}m}}s\\{u^2} = \frac{{q\sigma }}{{{\varepsilon _o}m}}s\end{aligned}\)

Substitute\(6.50 \times 1{0^{ - 9}}\,C\)for\(q\), \(5.90 \times 1{0^{ - 8}}\,C/{m^2}\)for\(\sigma \),\(0.3\,m\)for\(s\),\(8.85 \times 1{0^{ - 12}}\,F/m\)for\({\varepsilon _o}\) and\(8.20 \times 1{0^{ - 9}}\,kg\)for\(m\)into the above equation,

\(\begin{aligned}{u^2} = \frac{{\left( {6.50 \times 1{0^{ - 9}}\,C} \right)\left( {5.90 \times 1{0^{ - 8}}\,C/{m^2}} \right)}}{{\left( {8.85 \times 1{0^{ - 12}}\,F/m} \right)\left( {8.20 \times 1{0^{ - 9}}\,kg} \right)}}\left( {0.3\,m} \right)\\ = 1585.36\end{aligned}\)

\(\therefore u = 39.8\,m/s\)

Therefore the initial speed of the object is \(39.8\,m/s\).