A ball of radius R carries a positive charge whose volume density depends on a separation $r$ from the ball’s centre as $\rho ={\rho }_0(1-\frac{r}{R}),$ where ${\rho }_0$ is a constant. If the permittivities of the ball and the environment is equal to unity, then

A ball of radius R carries a positive charge with volume charge density:

ρ(r) = ρ₀(1 − r/R)

where ρ₀ is a constant. The permittivity inside and outside is the same (= ε₀). Options given:

• a) Electric field outside (r > R): ρ₀R³ / (12ε₀ r²)
• b) Electric field inside (r < R): (ρ₀ r / ε₀)(1 − 3r/4R)
• c) Maximum field E_max = ρ₀R / 9ε₀ at r = 2R/3
• d) Electric field inside (r < R): (ρ₀ r / ε₀)(1 + 3r/4R)

Step 1. Total charge

Q = ∫₀ᴿ ρ(r)·4πr²dr = (πρ₀R³)/3

Step 2. Field outside (r > R)

E = Q / (4π ε₀ r²) = ρ₀R³ / (12ε₀ r²)

Matches option (a).

Step 3. Field inside (r < R)

Charge enclosed up to r:

Q(r) = 4πρ₀ (r³/3 − r⁴/4R)

So electric field:

E(r) = (ρ₀r / ε₀)(1/3 − r/4R) = (ρ₀r / ε₀)(1 − 3r/4R)/3

Which matches option (b).

Step 4. Maximum field inside

Differentiating gives maximum at r = 2R/3. Value: E_max = ρ₀R / (9ε₀).

Matches option (c).

Option (d) has a plus sign instead of minus. That does not agree with the correct derivation. So (d) is wrong.

Correct options are: (a), (b), and (c).