two uniformly charged non-conducting spheres of radius r= 10 cm are placed on th

Question

two uniformly charged non-conducting spheres of radius r= 10 cm are placed on the horizontal axis as shown in the figure. The surface charge density on the left sphere is n1 = 3.5 ×10^-4 C/m^2 and on the right sphere is n^2= 1.5 ×10^-4 V/m^2. A negative point charge q=-2uC is placed on the vertical axis at 1 m above the

origin.

y (m) 2.0 O (m) -2.0-1.0 L0 2 a. Calculate the magnitude and direction of the net electric field at the origin (i.e. at coordinates (r.y) (0m, 1 m). c. Calculate the net force on the point charge q. d. What should the radius rz of the right sphere be in order for charge q to have a net force which points straight down? (Note that the radius ri of the left sphere is still 10em.)

Explanation / Answer

r = 10 cm = 0.1 m
n1 = 3.5 * 10^-4 C/m^2
n2 = 1.5 * 10^-4 C/m^2

q1 = 4**r^2 * n1
q1 = 4 * * 0.1^2 * 3.5 * 10^-4 C
q1 = 4.4 * 10^-5 C

q2 = 4**r^2 * n2
q2 = 4 * * 0.1^2 * 1.5 * 10^-4 C
q2 = 1.88 * 10^-5 C

Net Field at the origin,
E = k*q1/d^2 - k*q2/d^2
E = (8.9 * 10^9) / 2^2 * [4.4 * 10^-5 - 1.88 * 10^-5]
E = 56070 N/C
Direction :- Towards + ve X axis !!

(b)

d^2 = 1^2 + 2^2
d^2 = 5

Ex = k*q1/d^2 * 1/sqrt(5) + k*q2/d^2 * 1/sqrt(5)
Ex = (8.9 * 10^9 * (1/sqrt(5)) ) / 5 * [4.4 * 10^-5 + 1.88 * 10^-5]
Ex = 49991.3 N/C

Ey = k*q1/d^2 * 2/sqrt(5) - k*q2/d^2 * 2/sqrt(5)
Ey = (8.9 * 10^9 * (2/sqrt(5)) ) / 5 * [4.4 * 10^-5 - 1.88 * 10^-5]
Ey = 40120.4 N/C

(c)
E = sqrt(Ex^2 + Ey^2)
E = sqrt(49991.3^2 + 40120.4^2)
E = 64099.7 N/C

F = q*E
F = 2 * 10^-6 * 64099.7 N
F = 0.128 N

(d)
For that to happen,
q1 = q2
4**r2^2 * n1 = 4**r2^2 * n2
10^2 * 3.5 * 10^-4 = r2^2 * 1.5 * 10^-4
r2 = 15.3 cm

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