A system consists of a ball of radius R carrying a spherically symmetric charge and the surrounding space filled with a charge of volume density ρ = α/r, where α is a constant, r is the distance from the center of the ball. Find the ball’s charge for which the magnitude of the electric field strength vector is independent of r outside the ball. How high is this strength? The permittivity’s of the ball and the surrounding space are assumed to be equal to unity.
Text Solution
Verified by Expertsq = 2παR 2 , E = 
Sol. To calculate the electric field due to the charged sphere and the space surrounding the sphere, a shell of radius x and thickness dx whose center is the center of the sphere is taken and electric field due to this shell and charged sphere at a distance r from O is obtained as given below
E = 
× 
+ 
[where , q = charge considered on the ball]
Where, first term is the field strength of spherical charge q and second integral term is the field strength of space surrounding the charged sphere.
E = 
× 
+ 
so, E = 
+ 
or, = 
+ 
or, = 
+ 
or, = 
+ 
+ 
so, E = 
+ 
– 
Now, for E to be independent of r, sum of the first and third terms must be zero.
so, 
– 
= 0
or, q = 2παR 2 So, resultant field, independent of r, is given as
E = 
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