\(q_1, q_2,q_3~\text{and}~q_4\) are point charges located at points as shown in the figure and \(S\) is a spherical Gaussian surface of radius \(R\). Which of the following is true according to the Gauss’s law?

1. \(\oint_s\left(\vec{E}_1+\vec{E}_2+\vec{E}_3\right) \cdot d \vec{A}=\frac{q_1+q_2+q_3}{2 \varepsilon_0}\)
2. \(\oint_s\left(\vec{E}_1+\vec{E}_2+\vec{E}_3+\vec{E}_4\right) \cdot d \vec{A}=\frac{\left(q_1+q_2+q_3\right)}{\varepsilon_0}\)
3. \(\oint_s\left(\vec{E}_1+\vec{E}_2+\vec{E}_3\right) \cdot d \vec{A}=\frac{\left(q_1+q_2+q_3+q_4\right)}{\varepsilon_0}\)
4. \(\oint_s\left(\vec{E}_1+\vec{E}_2+\vec{E}_3+\vec{E}_4\right) \cdot d \vec{A}=\frac{\left(q_1+q_2+q_3+q_4\right)}{\varepsilon_0}\)

Consider the charge configuration and spherical Gaussian surface as shown in the figure. While calculating the flux of the electric field over the spherical surface, the electric field will be due to: 

(1) q₂ only

(2) Only the positive charges

(3) All the charges

(4) +q₁ and – q₁ only

The electric intensity due to an infinite cylinder of radius R and having charge q per unit length at a distance r(r > R) from its axis is 

(1) Directly proportional to r²

(2) Directly proportional to r³

(3) Inversely proportional to r

(4) Inversely proportional to r²

Two infinitely long parallel wires having linear charge densities λ₁ and λ₂ respectively are placed at a distance of R meters. The force per unit length on either wire will be $\left(K=\frac{{\displaystyle 1}}{{\displaystyle 4\pi {\epsilon }_0}}\right)$

(1) $K\frac{2{\lambda }_1{\lambda }_2}{R^2}$

(2) $K\frac{2{\lambda }_1{\lambda }_2}{R}$

(3) $K\frac{{\lambda }_1{\lambda }_2}{R^2}$ 

(4) $K\frac{{\lambda }_1{\lambda }_2}{R}$

Charge q is uniformly distributed over a thin half-ring of radius R. The electric field at the centre of the ring is 

(1) $\frac{q}{2{\pi }^2{\epsilon }_0{R}^2}$

(2) $\frac{q}{4{\pi }^2{\epsilon }_0{R}^2}$ 

(3) $\frac{q}{4\pi {\epsilon }_0{R}^2}$

(4) $\frac{q}{2\pi {\epsilon }_0{R}^2}$

Three positive charges of equal value q are placed at the vertices of an equilateral triangle. The resulting lines of force should be sketched as in 

(1)   (2)

(3)    (4)

An infinite number of electric charges each equal to \(5\) nC (magnitude) are placed along the \(x\text-\)axis at \(x=1\) cm, \(x=2\) cm, \(x=4\) cm, \(x=8\) cm ………. and so on. In the setup if the consecutive charges have opposite sign, then the electric field in Newton/Coulomb at \(x=0\) is: \(\left(\frac{1}{4 \pi \varepsilon_{0}} = 9 \times10^{9} ~\text{N-m}^{2}/\text{C}^{2}\right)\)
1. \(12\times 10^{4}\)
2. \(24\times 10^{4}\)
3. \(36\times 10^{4}\)
4. \(48\times 10^{4}\)

Two-point charges \(+q\) and \(–q\) are held fixed at \((–d, 0)\) and \((d, 0)\) respectively of a \((x, y)\) coordinate system. Then:

Four charges equal to – Q are placed at the four corners of a square and a charge q is at its centre. If the system is in equilibrium the value of q is 

1. $-\frac{Q}{4}(1+2\sqrt{2})$

2. $\frac{Q}{4}(1+2\sqrt{2})$

3. $-\frac{Q}{2}(1+2\sqrt{2})$

4. $\frac{Q}{2}(1+2\sqrt{2})$ 

The electric field due to a uniformly charged solid sphere of radius R as a function of the distance from its centre is represented graphically by -

(1)  (2)

(3)  (4)