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Question

Two non-conducting spheres of radii R1 and R2 carrying uniform volume charge densities +ρ andρ, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region:
31925.PNG
  1. the electrostatic field is zero
  2. the electrostatic potential is constant
  3. the electrostatic field is constant in magnitude
  4. the electrostatic field has same direction

A
the electrostatic field is constant in magnitude
B
the electrostatic field has same direction
C
the electrostatic field is zero
D
the electrostatic potential is constant
Solution
Verified by Toppr

Let the vector joining the centers of the two spheres be d
and consider a point P in the overlap region, whose position vector be r1 and r2 with respect to the centers of each sphere respectively.

Thus, by triangle law of vector addition, r1r2=d

Electric field at P due to the positively charged sphere is E1=14πϵ0qr31r1
Here, q is the total charge enclosed in the sphere of radius r1
Thus, q=ρ×43πr31
Hence, E1=ρ3ϵ0r1

Similarly, Electric field due to negatively charged sphere is E2=ρ3ϵ0r2

Thus, net electric field at P is E=E1+E2=ρ3ϵ0(r1r2)=ρ3ϵ0d

Both, direction and magnitude of the electric field are constant in the overlap region.

693503_31925_ans_101e86cf42674caa84616d9483311656.JPG

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