Question

The region between two concentric spheres of radii a < b contain volume charge density ρ(r) = `"c"/"r"`, where c is constant and r is radial- distanct from centre no figure needed. A point charge q is placed at the origin, r = 0. Value of c is in such a way for which the electric field in the region between the spheres is constant (i.e. independent of r). Find the value of c:

विकल्प

• `"q"/(2pi"a"^2)`

• `"q"/(4pi"a"^2)`

• `"q"/(pi"a"^2)`

• `"q"/"a"^2`

Answer 1

`bb("q"/(4pi"a"^2))`

Explanation:

Total flux = `"Total charge"/epsilon_0` (Gauss law)

`"E"xx4pir^2= "q"/epsilon_0+(4pi)/epsilon_0int_"a"^"r""c"/"r"xx"r"^2"dr"`

`"E"xx4pir^2= "q"/epsilon_0+(4pi"c"["r"^2-"a"^2])/epsilon_0`

E = `"q"/(4pi"r"^2epsilon_0)+("c"["r"^2-"a"^2])/("r"^2epsilon_0)`

E = `"q"/(4pi"r"^2epsilon_0)+"c"/(epsilon_0)-("ca"^2)/("r"^2epsilon_0)`

as E is independent of r

∴ `"q"/(4pi"r"^2epsilon_0)= ("ca"^2)/("r"^2epsilon_0)`

c = `"q"/(4pi"a"^2)`

Now E is `"q"/(4piepsilon_0"a"^2)`