Question

A spherically symmetric charge distribution is considered with charge density varying as
$ρ(r) = \begin{cases} ρ_0(\frac 34−\frac rR) &; \text{for} \ r≤R\\\ Zero &; \text{for}\ r>R \end{cases}$
Where, r(r<R) is the distance from the centre O (as shown in figure) The electric field at point P will be :

• A. $\frac {ρ_0r}{4ε_0}(\frac 34−\frac rR)$
• B. $\frac {ρ_0r}{3ε_0}(\frac 34−\frac rR)$
• C. $\frac {ρ_0r}{4ε_0}(1−\frac rR)$
• D. $\frac {ρ_0r}{5ε_0}(1−\frac rR)$

Answer 1

Answer: (C)

$\frac {ρ_0r}{4ε_0}(1−\frac rR)$

Answer 2

$(4πr^2)E_ρ=\frac {Q_{in}}{ε_0}$

$=\frac {∫_0^r ρ_0(\frac 34−\frac rR)4πr^2dr}{ε_0}$

$=\frac {ρ_0\pi4}{ε_0}(\frac {r^3}{4}−\frac {r^4}{4R})$
Now, the electric field at point P will be:

$E_ρ=\frac {ρ_0}{4ε_0}(r−\frac {r^2}{R})$

$E_ρ=\frac {ρ_0r}{4ε_0}(1−\frac rR)$

So, the correct option is (C): $\frac {ρ_0r}{4ε_0}(1−\frac rR)$