Begin2.DVI

9-42. Couloumb’s law states that the force between two charges q 1 and q 2 acts

along the line joining the two charges and the magnitude of the force varies directly

to the product of charges and inversely as the square of the distance rbetween the

charges. Symbolically the magnitude of this force is F =q 1 q 2 /r^2 in the appropriate

system of units^15. A charge Qis called a test charge if it is located at a variable point

(x, y, z )and experiences a force from another charge q 1 ,located at a fixed point. The

ratio of the force experienced by the test charge to the magnitude of the test charge

is called the electrostatic intensity E
at the point (x, y, z )and is given by E
=FQ.An

equivalent statement is that a unit charge has been placed at the point P and the

electrostatic intensity is the total force which acts on this test charge.

(a) Show that a charge qlocated at the origin produces an electrostatic intensity E

at a point (x, y, z )given by

E
 =q
r

r^3 , where r=|
r | and
r =x

ˆe 1 +yˆe 2 +zˆe 3.

(b) Show that (i) E
=−∇

q

r (ii)

E
has the scalar potential φ=q

r

(iii) curlE
= 0 and (iv) div E
=−q∇^2

1

r= 0

c) Let q 1 denote a fixed charge located at the point P 1 (x 1 , y 1 , z 1 )and let q 2 denote

another fixed charge located at the point P 2 (x 2 , y 2 , z 2 ). Show the electrostatic

intensity on a test charge at (x, y, z)is given by

E
 =−q^1 
r^1

r 12

−q^2 
r^2

r^32

,

where ri=|ri|for i= 1, 2 ,and
r i= (xi−x)ˆe 1 + (yi−y)ˆe 2 + (zi−z)ˆe 3.

(d) Show for n charges qi, i = 1, 2 ,... , n, located respectively at the points Pi(xi, y i, zi),

for i= 1, 2 ,... , n , the electrostatic intensity at a general point (x, y, z )is given by

E
 =−

∑n

i=1

qi

r^3 i

ri,

where
ri= (xi−x)ˆe 1 + (yi−y)ˆe 2 + (zi−z)ˆe 3 and ri=|ri|for i= 1, 2 ,.. ., n.

(^15) If charges are measured in units of statcoulombs and distance is measured in centimeters, then the force has
units of dynes.