A Charged Isolated Conductor — Question 3

Question 3

Separate cavities impose separate induced-charge constraints

A connected isolated conductor of net charge $+5.00$ nC contains two closed cavities. Their isolated contents have net charges $q_1=+4.00$ nC and $q_2=-6.00$ nC. No contents touch the metal.

1. Find the induced charge on each cavity wall and on the outer surface.

2. An external insulating mechanism adds $+2.00$ nC to the second cavity contents without transferring charge to the conductor. Find all three new surface totals and their changes.

Question 3 - Solution

Separate cavities impose separate induced-charge constraints

(a) Method and calculation

$$Q_{w1}=-q_1=-4.00\,\mathrm{nC},\quad Q_{w2}=-q_2=+6.00\,\mathrm{nC}.$$$$Q_o=Q_c+q_1+q_2=+3.00\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$q_2\prime=-4.00\,\mathrm{nC}\Rightarrow Q_{w1}\prime=-4.00,\quad Q_{w2}\prime=+4.00,\quad Q_o\prime=+5.00\quad(\mathrm{nC}).$$$$\Delta Q_{w1}=0,\quad\Delta Q_{w2}=-2.00,\quad\Delta Q_o=+2.00\quad(\mathrm{nC}).$$The metal redistributes charge while retaining its specified net total.

Physical check

The two separate Gaussian surfaces in the metal each enforce zero enclosed charge around their respective cavity.