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

The Correct Option is C

\((4πr^2)E_ρ=\frac {Q_{in}}{ε_0}\)

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

\(=\frac {ρ_0\pi4}{ε_0}(\frac {r^3}{4}−\frac {r^4}{4R})\)
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): \(\frac {ρ_0r}{4ε_0}(1−\frac rR)\)