(a) Charge placed at the centre of the hollow sphere is −q. Hence, a charge of magnitude +q will be induced to the inner surface. Therefore, total charge on the inner surface of the shell is +q. Surface charge density at the inner surface
σi=‌

‌ Total charge ‌

‌ Inner surface area ‌

=‌

+q

4πr12

A charge of −q is induced on the outer surface of the sphere. A charge of magnitude Q is placed on the outer surface of the sphere. Therefore, total charge on the outer surface of the sphere is Q−q. Surface charge density at the outer surface
σ‌outer ‌=‌

‌ Total charge ‌

‌ Outer surface area ‌

=‌

Q−q

4πr22

(b) Electric field at point lying outside the sphere at a distance r from the centre of the sphere:
Applying Gauss theorem
‌ Flux ‌=ϕ=‌

‌ Charge enclosed ‌

εo

‌ Or, ‌ E×4πr2‌=‌

Q−q

εn

∴E‌=‌

Q−q

4πr2ε0

OR
(a) Electric field due to an infinitely long straight wire having uniform linear charge density λ : x= distance of the point P from the wire where the electric field is to be evaluated
E= electric field at the point P
A Gaussian cylinder of length l, radius x is considered.
An infinitesimally small area ds on the Gaussian surface is considered.
Electric field is same at all points on the curved surface of the cylinder and directed radially outward. So, E and ds are along the same direction.
The total electric flux (ϕ) through curved surface =∫Eds‌cos‌θ
Since E and ds are along the same direction, so θ= 0∘
So, ‌‌ϕ=E(2πxl)
The net charge enclosed by Gaussian surface is, q= λl
∴ By Gauss's law,
ϕ=‌

1

ε0

q
Or, ∴‌

1

ε0

q=E(2xrl)
∴E=‌

λ

2πxε0

(b)