Question

Question: (II) In Fig. 16–62, two objects, \({{\bf{O}}_{\bf{1}}}\) and \({{\bf{O}}_{\bf{2}}}\) have charges \({\bf{ + 1}}{\bf{.0}}\;{\bf{\mu C}}\) and \({\bf{ - 2}}{\bf{.0}}\;{\bf{\mu C}}\), respectively, and a third object, \({{\bf{O}}_{\bf{3}}}\), is electrically neutral. (a) What is the electric flux through the surface \({A_1}\) that encloses all three objects? (b) What is the electric flux through the surface \({A_2}\) that encloses the third object only?

FIGURE 16–62 Problem 39.

Short answer

(a) The electric flux through surface \({A_1}\) that encloses all three objects is

(b) The electric flux through surface \({A_2}\) that encloses the third object only is \(0\).

Step-by-step solution

Determination of the electric flux value

The value of the electric flux can be obtained by dividing the value of the net charge by the value of the permittivity of free space.

Given information

Given data:

The charge on object \({{\rm{O}}_{\rm{1}}}\) is \({q_1} = + 1.0\;{\rm{\mu C}}\).

The charge on object \({{\rm{O}}_{\rm{2}}}\) is \({q_2} = - 2.0\;{\rm{\mu C}}\).

The charge on object \({{\rm{O}}_{\rm{3}}}\) is \({q_3} = 0\).

Evaluation of the electric flux through surface \({A_1}\) that encloses all three objects

(a)

The net charge enclosed within surface \({A_1}\) can be calculated as:

\(\begin{aligned}{c}{q_{{\rm{net}}}} &= {q_1} + {q_2} + {q_3}\\{q_{{\rm{net}}}} &= \left( { + 1.0\;{\rm{\mu C}}} \right) + \left( { - 2.0\;{\rm{\mu C}}} \right) + 0\\{q_{{\rm{net}}}} &= - 1.0\;{\rm{\mu C}}\end{aligned}\)

The electric flux through surface \({A_1}\) that encloses all three objects can be calculated as:

Thus, the electric flux through surface \({A_1}\) that encloses all three objects is

Evaluation of the electric flux through surface \({A_2}\) that encloses the third object only

(b)

The net charge enclosed within surface \({A_2}\) is equal to the charge on object \({{\rm{O}}_{\rm{3}}}\). Therefore, the net charge enclosed within the surface \({A_2}\) is \({q_{{\rm{net}}}} = 0\).

So, the electric flux through surface \({A_2}\) can be calculated as:

\(\begin{aligned}{l}{\phi _{\rm{E}}} &= \frac{{{q_{{\rm{net}}}}}}{{{\varepsilon _{\rm{o}}}}}\\{\phi _{\rm{E}}} = \frac{0}{{{\varepsilon _o}}}\\{\phi _{\rm{E}}} &= 0\end{aligned}\)

Thus, the electric flux through surface \({A_2}\) that encloses the third object only is \(0\).