Abhimanyu Kumar

(b) After the supply was disconnected.

We know, capacitance = dielectric constant × C₀
C = KC₀ = 6 × 18pF = 108pF [ see question - 2.8]
(a) if the voltage supply remained connected , then the potential difference across the capacitor will remain the same. e.g., V = 100V and hence, charge on the capacitor becomes Q = CV = 108pF × 100V
Q = 108 × 10⁻¹² × 100 = 1.08 × 10⁻⁸ C

(b) if the voltage supply was disconnected , then charge on the capacitor reamins the same. e.g., Q = 1.8 × 10⁻⁹C [ see question - 2.8]
And hence the potential difference across the capacitor becomes
V = Q/C = 1.8 × 10⁻⁹/1.08 × 10⁻¹⁰ = 16.6V

A regular hexagon ABCDEF of side 10cm has a charge 5 μC at each of its vertices as shown in the figure.
Potential due to charge placed at A, at the centre is Kq/r
Potential due to charge placed at B, at the centre is Kq/r

Potential due to Charge placed at F , at centre is Kq/r
so, total potential at centre of regular hexagon , V = V₁ + V₂ + V₃ + V₄ + V₅ + V₆
so, V = kq/r + kq/r + kq/r + kq/r + kq/r + Kq/r
V = 6kq/r
Hence, k = 9 × 10⁹Nm²/C² , q = 5 × 10⁻⁶C and r = 10cm = 0.1 m
Now, V = 6 × 9 × 10⁹ × 5 × 10⁻⁶/0.1 = 2.7 × 10⁶ Volts.

initially d,

therefore, C= EA/d=8 -(1)

now d1=d/2, and dielectric is put,

therefore C1=KEA/d1. -(2) here k=6 which is dielectric constant

we know d1=d/2 substituting it in (1)

C1=6EA/(d/2)=12EA/d. -(3)

(3)÷(1)

C1÷8=12EA/d÷EA/d

C1=96 uF

(c) At a point 18 cm from the centre of the sphere?

a) We know, charge have nature to reside outer surface of the conductor. It means, charge inside the surface equals zero.
According to Gaussian theorem,
Ф = q/ε ₀, here q is charged inclosed the Gaussian surface.
∵ q = 0
so, Ф = 0 and flux , Ф = E.A = 0
so, E = 0
Hence, inside the sphere, the electric field equals zero.
(b) Take a Gaussian surface of radius r > R = 12cm
then, charged inclosed into the Gaussian surface is q = 1.6 × 10⁻⁷ C
So, Ф = q/ε₀
So, EA = q/ε₀
E = q/ε₀A, here A is the surface area of Gaussian spherical surface
e.g., A = 4πr²
So, E = q/4πε₀r² = 9 × 10⁹ × 1.6 × 10⁻⁷/(12 × 10⁻²)²
= 10⁵ N/C
(C) Similarly explanation of (B),
So, E = kq/r²
Here , k = 9 × 10⁹ Nm²/C² , q = 1.6 × 10⁻⁷C and r = 18cm
So, E = 9 × 10⁹ × 1.6 × 10⁻⁷/(18 × 10⁻²)²
= 4.44 × 10⁴ N/C.