(i) At a distance r; the force on the small sphere due to large sphere
\(F=\frac { 1 }{ 4\pi { \varepsilon }_{ 0 } } .\frac { Qq }{ { mr }^{ 2 } } \)
(ii) If m is the mass of small sphere then its acceleration
\(a=\frac { F }{ m } =\frac { 1 }{ 4\pi { \varepsilon }_{ 0 } } .\frac { Qq }{ { mr }^{ 2 } } \)
(iii) As the small sphere is pushed away (i.e. r increased) 'a' decrease.
(iv) As 'a' is always +ve the speed of the small sphere goes on increasing.
(v) ∴ increasing velocity and decreasing acceleration.
(d) is correct

(i) The dipole moment of dipole \(|\overset { \rightarrow }{ P } |=q\times 2a\)
Force on -q at A = -q \(\overset { \rightarrow }{ E } \)
Force on + q at B = +q\(\overset { \rightarrow }{ E } \)
Net force on the dipole = q\(\overset { \rightarrow }{ E } \)- q\(\overset { \rightarrow }{ E } \) = 0
(ii) Work done on dipole when it is rotated through 180°
W = ∆U = pE (cos θ₁ - cos θ₂)
= pE (cos 0° - cos180^o)
= pE(1 - (-1)
W = 2pE

(i) In electric field line is the path of movement of a positive test charge (q₀ ⇾ 0) A moving charge experiences a continuous force is an electric field, so field line is always continuous
(ii) The field lines nevel intersect since if they cross, there will be two directions of electric field at the point of intersection, which is impossible.

(i) In stable equilibrium the dipole moment is parallel to he direction of electric field. i.e. θ = 0.
 θ = 0^o \(\overset { \rightarrow }{ P } \) is parallel to stable \(\overset { \rightarrow }{ E } \) equilibrium
(ii) In unstable equilibrium, P.E. is max., so θ = π. i.e. dipole moment is antiparallel to electric field.
θ = 180°. \(\overset { \rightarrow }{ P } \) is antiparallel

(i) Electric flux through one face \(=\frac { 1 }{ 6 } .\frac { Q }{ { \varepsilon }_{ 0 } } \mu { V }_{ m }\)
(ii) By symmetry the flux through each of the six faces of cube will be same when charge is placed at the centre.
\(\\ \therefore { \phi }_{ E }=\frac { 1 }{ 6 } .\frac { Q }{ { \varepsilon }_{ 0 } } \)
Thus electric flux passing through two opposite faces of the cube
\(=2\times \frac { 1 }{ 6 } .\frac { Q }{ { \varepsilon }_{ 0 } } \)
\(\phi =\frac { 1 }{ 3 } .\frac { Q }{ { \varepsilon }_{ 0 } } \)

Net electric flux is zero because
(i) It is independent to the shape and size
(ii) Net charge of the electric dipole is zero.

(i) Charge on each sphere \(=\frac { { q }_{ A }-{ 3q }_{ A } }{ 2 } =-{ q }_{ A }\)
(ii)New charge on A is \(\frac { { q }_{ A } }{ 2 } \)
New charge on B is \(\frac { { q }_{ A }(2{ q }_{ B }) }{ 4 } \)
\(\because \frac { \frac { { q }_{ A } }{ 2 } +{ q }_{ B } }{ 2 } =\frac { { q }_{ A } }{ 4 } +\frac { q_{ B } }{ 2 } =\frac { { q }_{ A }+2{ q }_{ B } }{ 4 } \)
q_B = -3q_A
∵ q_B = -3q_A
∴ New charge on B₁ is \(\frac { { q }_{ A }-{ 6q }_{ A } }{ 4 } \)
New charge on B \(-\frac { 5 }{ 4 } { q }_{ A }\)

To rotate any object, there must be a torque applied on the object. For the ceiling fan, the initial torque is given by the capacitor widely known as a condenser. If the condenser is faulty, it will not give sufficient initial torque to rotate the blades when the fan is switched on.

Capacitor is a device used to store electric charge and electrical energy.

When the external electric field applied to a dielectric is very large, it tears the atoms apart so that the bound charges become free charges. Then the dielectric starts to conduct electricity. This is called dielectric breakdown.

When an external electric field is applied, the dipoles inside the polar molecule tend to align in the direction of the electric field. Hence a net dipole moment is induced in it. Then the dielectric is said to be polarized by an external electric field.

(i) In polar molecules, the centers of the positive and negative charges are separated even in the absence of an external electric field.
(ii) They have a permanent dipole moment.
(iii) Examples : H₂O, N₂O, HCI and NH₃.

When an external electric field is applied, the centers of positive and negative charges are separated by a small distance which induces dipole moment in the direction of the external electric field. Then the dielectric is said to be polarized by an external electric field.

(i) A non-polar molecule is one in which centers of positive and negative charges coincide. As a result, it has no permanent dipole moment.
(ii) Examples of non-polar molecules are hydrogen (H₂), oxygen (O₂), and carbon di oxide (CO₂).

(i) Both forces obey inverse square law, F∝\(\frac{1}{r^2}\)
(ii) Both forces are proportional to product of masses or charges.
(iii) Both forces are conservative forces.
(iv) Both forces can operate in vacuum.