# Solved problems

## 4. Capacity

### Problem 2

(a) Determine the capacity of an isolated conducting sphere with radius of 4.0 cm and surrounded by air. (b) The sphere is cover by a layer of insulating material, with a thickness of 1 mm and a dielectric constant of 5.6, and a second metal sphere with radius of 4.1 cm is placed over the insulator, thus forming a spherical capacitor. Determine the capacity of this capacitor. (c) What is the ratio between the capacities of the condenser and the sphere?

(a) The capacity of the sphere is:

(b) The capacity of the spherical capacitor is:

(c) The ratio between the two capacities is

### Problem 3

The three capacitors in the figure have capacities $C_1$ = 1.2 μF, $C_2$ = 4.3 μF and $C_3$= 2.5 μF. The voltage between points A and B is 9.0 V. (a) Compute the charge stored in each capacitor. (b) Compute the total energy stored in the system.

Capacitors $C_1$ and $C_2$, in parallel, can be replaced by $C_4$ = $C_1$ + $C_2$= 5.5 μF. The equivalent capacitor between A and B is$C_5$ = $C_4\,C_3/(C_4+C_3)$ = 1.71875 μF.

(a) The charge stored in $C_5$ It is $Q_5$ = $C_5\,\varepsilon$ = 15:47 μC, which is also the charge stored on the capacitors $C_3$ and $C_4$. The potential difference$C_4$ It is $V_4$ = $Q_4/C_4$ = 2.8125 V and charges the first two condensers are $Q_1$ = $C_1\,V_4$ = 3.375 and μC $Q_2$ = $C_2\,V_4$ = 9.12 μC.

(b) The energy stored in the system is the same energy stored in the equivalent capacitor:$U_5$ = $C_5\,\varepsilon^2/2$ = 69.61 mJ.

### Problem 4

A plane-parallel plates capacitor with plates of 12 cm2 of area, separated by 1 cm, is completely filled by two dielectrics, each with a thickness of 0.5 cm and area equal to the area of the plates. Compute the capacity of the condenser knowing that the constants of the dielectrics are 4.9 and 5.6 (hint: assume that the capacitor is equivalent to two capacitors in series, each with a different dielectric).

The capacities of the two capacitors are

and the capacity of the system in series is:

### Problem 7

In the circuit of the figure, calculate the equivalent capacity: ( a ) Between points B and D. ( b ) between points A and B.

( A ) Among the B and D, the equivalent capacity is:

( B ) Between points A and B,