6 CIRCUIT CONCEPTS

(c) If another chargeQ 3 =− 2 μC is added to the system 1 m aboveQ 2 , as shown in Figure

E1.1.1, calculate the force exerted onQ 2.

Solution

(a) In the region where both ions and free electrons exist, their opposite charges cancel, and

the net charge density is zero. From the region containing ions only, the volume-charge

density is given by

ρ=( 1025 )( 1. 6 × 10 −^19 )= 1. 6 × 106 C/m^3

The net total charge is then calculated as:

Q=ρv=( 1. 6 × 106 )( 1. 25 × 10 −^6 × 10 −^3 × 10 −^3 )= 2 × 10 −^6 C

(b) The rectangular coordinate system shown defines the locations of the charges:Q 1 =

2 × 10 −^6 C;Q 2 = 3 × 10 −^6 C. The force thatQ 1 exerts onQ 2 is in the positive direction

ofx, given by Equation (1.1.1),

F ̄ 12 =(^3 ×^10

− (^6) )( 2 × 10 − (^6) )
4 π( 10 −^9 / 36 π) 22
a ̄x= ̄ax 13. 5 × 10 −^3 N
This is the force experienced byQ 2 due to the effect of the electric field ofQ 1. Note the
value used for free-space permittivity,ε 0 ,as(8. 854 × 10 −^12 ), or approximately 10−^9 / 36 π
F/m.a ̄xis the unit vector in the positivex-direction.
(c) WhenQ 3 is added to the system, as shown in Figure E1.1.1, an additional force onQ 2
directed in the positivey-direction occurs (sinceQ 3 andQ 2 are of opposite sign),
F ̄ 32 =(^3 ×^10
− (^6) )(− 2 × 10 − (^6) )
4 π( 10 −^936 π) 12
(− ̄ay)= ̄ay 54 × 10 −^3 N
The resultant forceF ̄ 2 acting onQ 2 is the superposition ofF ̄ 12 andF ̄ 32 due toQ 1 and
Q 3 , respectively.
The vector combination ofF ̄ 12 andF ̄ 32 is given by:
F ̄ 2 =
√
F 122 +F 322  tan−^1
F ̄ 32
F ̄ 12

√
13. 52 + 542 × 10 −^3  tan−^1
54
13. 5
= 55. 7 × 10 −^3  76° N

Conductors and Insulators

In order to put charge in motion so that it becomes an electric current, one must provide a path

through which it can flow easily by the movement of electrons. Materials through which charge

flows readily are calledconductors. Examples include most metals, such as silver, gold, copper,

and aluminum. Copper is used extensively for the conductive paths on electric circuit boards and

for the fabrication of electrical wires.