Question

Figure 21-20 shows three identical conducting bubblesA,B, andCfloating in a conducting container that is grounded by a wire. The bubbles initially have the same charge. BubbleAbumps into the container’s ceiling and then into bubbleB. Then bubbleBbumps into bubbleC, which then drifts to the container’s floor. When bubbleCreaches the floor, a charge ofis transferred upward through the wire, from the ground to the container, as indicated. (a) What was the initial charge of each bubble? When (b) bubbleAand (c) bubbleBreach the floor, what is the charge transfer through the wire? (d) During this whole process, what is the total charge transfer through the wire?

Short answer

• a)The initial charge of each bubble is.
• b)When bubble A reaches the floor, the charge transfer through the wire is -2e.
• c)When bubble B reaches the floor, the charge transfer through the wire is- 3e.
• d)During this whole process, the total charge transfer through the wire is -12e.

Step-by-step solution

 Stating given data

• a) Figure 21-20 shows three identical conducting bubbles A, B, and Cfloating in a conducting container that is grounded by a wire.
• b)The bubbles havethesame charge.

c) When bubble C reaches the floor, the charge transfer upward through the wire is -3e

Understanding  concept of charging by conduction

When two charged bodies are brought in contact with each other, the total charge of the system is divided among the given charged bodies equally untiltheyhave the same charge as per their volume. This process is called charging by conduction. Thus, using this concept, the charges on each body can be calculated.

a) Calculation of initial charge of each bubble 

Letthe charge on each bubble initially be q.

At first, when bubble A comes in contact with the container surface, it becomes neutral with charge $q_A=0$.

Now, after it comes in contact with bubble B, their charges get equally distributed as per the concept. Thus, the new charge on A and B becomes

$$\begin{gathered} q_A\text{'}=q_B\text{'} \\ =\frac{0+q}{2} \\ =\frac{q}{2} \end{gathered}$$

Now, after bubble B comes in contact with bubble C, their charges get equally distributed as per the concept. Thus, the new charge on B and C becomes

localid="1663082080063" $$\begin{gathered} q_B\text{'}\text{'}=q_c\text{'} \\ =\frac{{\displaystyle \frac{q}{2}}+q}{2} \\ =\frac{3q}{4} \end{gathered}$$

Thus, the final charges become$q_A\text{'}=q/2,q_B\text{'}\text{'}=3q/4,q_C\text{'}=3q/4$.

So, as per the given data, the charge is being transferred through the wire intheupward direction for bubble C reaching the floor. Thus, as per the concept, any charge in contact withtheground becomes neutral and sothenet charge on C should be zero after final transfer.

So, the final charge on C that is transferred by the bubble is equal to the negative value -3e for the transfer to happen. That implies

3q / 4= -(-3e)
q=4e

Hence, the initial charge is 4e.

b) Calculation of charge transferred by A when it reaches floor  

When A reachesthefloor,it transfers all itscharges to the ground, Thus, the net charge transferredisnegative for the flow of charge through the wire to happen.

That implies

$$\begin{gathered} q_A\text{'}=-(q/2) \\ =-(4e/2) \\ =-2e \end{gathered}$$

Hence, the net charge transferred by A is -2e.

c) Calculation of charge transferred by B when it reaches floor  

When B reachesthefloor,it transfers all itscharges to the ground. Thus, the net charge transferredisnegative for the flow of charge through the wire to happen.

That implies

$$\begin{gathered} q_B\text{'}\text{'}=-(3q/4) \\ =-\left(3\right(4e)/4) \\ =-3e \end{gathered}$$

Hence, the net charge transferred by A is -3e .

d) Calculation of net charge transferred by all bubbles 

The net charge transferred by all the bubbles during the whole process is given by

$$\begin{gathered} q^{\text{'}}=-\left(3q\right) \\ =-3\left(4e\right) \\ =-12e \end{gathered}$$

Hence, the total charge transferred is -12e .