Q1: Two point charges separated by distance $\displaystyle {r}$ in air exert a force $\displaystyle {F}$. The distance $\displaystyle {r'}$ at which these charges exert the same force in a medium of dielectric constant $\displaystyle {K}$ is:

(a) $\displaystyle {r}$     (b) $\displaystyle {r/K}$     (c) $\displaystyle {{\left( {\frac{{r^{2}}}{{K}}} \right)}^{{{1}/{2}\;}}} = {\frac{{r}}{{\sqrt{K}}}}$     (d) $\displaystyle {r\sqrt{K}}$

Q2: Charge $\displaystyle {{q}_{2}}$ of mass $\displaystyle {m}$ revolves around stationary charge $\displaystyle {{q}_{1}}$ in a circular orbit of radius $\displaystyle {r}$. The orbital periodic time of $\displaystyle {{q}_{2}}$ is:

(a) $\displaystyle {{\left( {\frac{{4{{\pi }^{3}}m{{r}^{2}}}}{{k{{q}_{1}}{{q}_{2}}}}} \right)}^{{{1}/{2}\;}}}$     (b) $\displaystyle {{\left( {\frac{{k{{q}_{1}}{{q}_{2}}}}{{4{{\pi }^{2}}m{{r}^{2}}}}} \right)}^{{{1}/{2}\;}}}$
(c) $\displaystyle {{\left( {\frac{{4{{\pi }^{2}}m{{r}^{4}}}}{{k{{q}_{1}}{{q}_{2}}}}} \right)}^{{{1}/{2}\;}}}$     (d) $\displaystyle {{\left( {\frac{{4{{\pi }^{2}}m{{r}^{3}}}}{{k{{q}_{1}}{{q}_{2}}}}} \right)}^{{{1}/{2}\;}}}$

Q3: Two small spheres each having charge $\displaystyle {+Q}$ are suspended by insulating threads of length $\displaystyle {L}$. In space with no gravitational effect, the angle between threads and tension in each thread will be:

(a) $\displaystyle {180^\circ, {\frac{{1}}{{4{\pi}{{\varepsilon}_{0}}}}}{\frac{{{Q}^{2}}}{{{\left( {2L} \right)}^{2}}}}}$     (b) $\displaystyle {90^\circ, {\frac{{1}}{{4{\pi}{{\varepsilon}_{0}}}}}{\frac{{{Q}^{2}}}{{{L}^{2}}}}}$
(c) $\displaystyle {180^\circ, {\frac{{1}}{{4{\pi}{{\varepsilon}_{0}}}}}{\frac{{{Q}^{2}}}{{2{L}^{2}}}}}$     (d) $\displaystyle {180^\circ, {\frac{{1}}{{4{\pi}{{\varepsilon}_{0}}}}}{\frac{{{Q}^{2}}}{{{L}^{2}}}}}$

Q4: Dimensional formula for the permittivity of free space $\displaystyle {{\varepsilon}_{0}}$ is:

(a) $\displaystyle {{\left[ {M^{-1}L^{-3}T^{4}A} \right]}}$     (b) $\displaystyle {{\left[ {M^{-1}L^{-3}T^{-3}A^{2}} \right]}}$     (c) $\displaystyle {{\left[ {M^{-1}L^{-3}T^{4}A^{2}} \right]}}$     (d) $\displaystyle {{\left[ {M^{-1}L^{-3}T^{4}A^{3}} \right]}}$

Assertion (A): When bodies are charged through friction, charge is transferred from one body to another, but no creation or destruction of charge occurs.
Reason (R): This follows from the law of conservation of electric charges.

Assertion (A): If an electric dipole of dipole moment $\displaystyle {30 \times 10^{-5}\text{ C}\cdot\text{m}}$ is enclosed by a closed surface, net flux coming out of the surface is zero.
Reason (R): An electric dipole consists of two equal and opposite charges.

Q1: Why do two electric field lines never cross each other?

Ans: If two lines intersect at a point, two tangents could be drawn at that intersection, representing two different directions of electric field at a single point, which is physically impossible.

Q2: What does $\displaystyle {{q}_{1} + {q}_{2} = 0}$ signify in electrostatics?

Ans: It signifies that electric charges are additive and $\displaystyle {{q}_{1}}$ and $\displaystyle {{q}_{2}}$ are equal in magnitude but opposite in sign ($\displaystyle {{q}_{1} = -{q}_{2}}$).

Q3: How does electrostatic force between two charges change when a plastic sheet is inserted between them?

Ans: Since dielectric constant for plastic $\displaystyle {K > 1}$, the electrostatic force decreases according to $\displaystyle {{F_{m} = {\frac{{F_{\text{vacuum}}}}{{K}}}}}$

Q1: Using Gauss's law, derive an expression for the electric field intensity due to an infinitely long straight wire of linear charge density $\displaystyle {\lambda}$.

Derivation:

1. Consider a coaxial cylindrical Gaussian surface of radius $\displaystyle {r}$ and length $\displaystyle {l}$ around the wire. Total charge enclosed is $\displaystyle {q = \lambda l}$.

2. Flux passes only through curved cylindrical surface as $\displaystyle {{\vec{E}} \perp d{\vec{A}}}$ at end caps.

3. By Gauss's Law:

$\displaystyle {\oint {{\vec{E}} \cdot d{\vec{A}}} = {\frac{{q}}{{{\varepsilon}_{0}}}} \implies E {\left( {2{\pi} r l} \right)} = {\frac{{\lambda l}}{{{\varepsilon}_{0}}}} \implies E = {\frac{{\lambda}}{{2{\pi}{{\varepsilon}_{0}}r}}}}$

Q2: Derive electric field intensity due to a thin spherical shell of radius $\displaystyle {R}$ at (i) outside point ($\displaystyle {r > R}$) and (ii) inside point ($\displaystyle {r < R}$). Plot $\displaystyle {E}$ vs $\displaystyle {r}$.

Derivation:

(i) Outside ($\displaystyle {r > R}$): Construct spherical Gaussian surface of radius $\displaystyle {r}$. Enclosed charge is $\displaystyle {q}$.

$\displaystyle {\oint {{\vec{E}} \cdot d{\vec{A}}} = E {\left( {4{\pi} r^{2}} \right)} = {\frac{{q}}{{{\varepsilon}_{0}}}} \implies E = {\frac{{1}}{{4{\pi}{{\varepsilon}_{0}}}}}{\frac{{q}}{{{r}^{2}}}}}$

(ii) Inside ($\displaystyle {r < R}$): Charge inside shell is zero ($\displaystyle {q_{\text{enclosed}} = 0}$).

$\displaystyle {E {\left( {4{\pi} r^{2}} \right)} = 0 \implies E = 0}$

Passage: Surface charge density is defined as charge per unit area $\displaystyle {\sigma = {\frac{{dq}}{{dS}}}}$. Two large thin metal plates A and B are placed parallel and close to each other with opposite surface charge densities of magnitude $\displaystyle {17.0 \times 10^{-22}\text{ C/m}^{2}}$.

Q1: What is the electric field $\displaystyle {E}$ in the outer region of the first plate?
Ans: Zero ($\displaystyle {E_{I} = {\frac{{\sigma}}{{2{{\varepsilon}_{0}}}}} - {\frac{{\sigma}}{{2{{\varepsilon}_{0}}}}} = 0}$).

Q2: What is the electric field $\displaystyle {E}$ between the two plates?
Ans:
$\displaystyle {E = {\frac{{\sigma}}{{{\varepsilon}_{0}}}} = {\frac{{17.0 \times 10^{-22}}}{{8.854 \times 10^{-12}}}} = 1.9 \times 10^{-10}\text{ N/C}}$

Q3: What is the ratio of electric field at distances 2 cm and 4 cm from plate A between the plates?
Ans: $\displaystyle {1 : 1}$ (The electric field due to infinite plane sheets is uniform and independent of distance).