In a region, the potential is represented by \(V=(x,y,z)=6x-8xy-8y+6yz,\) where \(V\) is in volts and \(x,y,z\) are in meters. The electric force experienced by a charge of \(2\) coulomb situated at a point \((1,1,1)\) is:
1. \(6\sqrt{5}~\text{N}\)
2. \(30~\text{N}\)
3. \(24~\text{N}\)
4. \(4\sqrt{35}~\text{N}\)

The grid (each square of 1m × 1m), represents a region in space containing a uniform electric field.

If potentials at points O, A, B, C, D, E, F and G, H are respectively 0, –1, –2, 1, 2, 0, –1, 1 and 0 volts, find the electric field intensity –

1. $\left(\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)<mspace\; width="1em"></mspace>\mathrm{V}/\mathrm{m}$                           

2. $\left(\hat{\mathrm{i}}-\hat{\mathrm{j}}\right)<mspace\; width="1em"></mspace>\mathrm{V}/\mathrm{m}$

3. $\left(-\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)<mspace\; width="1em"></mspace>\mathrm{V}/\mathrm{m}$                         

4. $\left(-\hat{\mathrm{i}}-\hat{\mathrm{j}}\right)<mspace\; width="1em"></mspace>\mathrm{V}/\mathrm{m}$

An electric field is given by ${\mathrm{E}}_{\mathrm{x}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>–<mspace\; width="1em"></mspace>2{\mathrm{x}}^3<mspace\; width="1em"></mspace>\mathrm{kN}/\mathrm{C}$. The potential of the point (1, –2), if potential of the point (2, 4) is taken as zero, is –

1.  $-7.5\times {10}^3<mspace\; width="1em"></mspace>\mathrm{V}$                     

2.  $7.5\times {10}^3<mspace\; width="1em"></mspace>\mathrm{V}$

3.  $-15\times {10}^3<mspace\; width="1em"></mspace>\mathrm{V}$                     

4.  $15\times {10}^3<mspace\; width="1em"></mspace>\mathrm{V}$

A network of four capacitors of capacity equal to $C_1=C,C_2=2C,C_3=3C$ and C₄ = 4 C are connected in a battery as shown in the figure. The ratio of the charges on C₂ and C₄ is 

(1) $\frac{22}{3}$

(2) $\frac{3}{22}$

(3) $\frac{7}{4}$

(4) $\frac{4}{7}$

In the given network capacitance, C₁ = 10 μF, C₂ = 5 μF and C₃ = 4 μF. What is the resultant capacitance between A and B 

1. 2.2 μF

2. 3.2 μF

3. 1.2 μF

4. 4.7 μF

Two condensers C₁ and C₂ in a circuit are joined as shown in figure. The potential of point A is V₁ and that of B is V₂. The potential of point D will be 

1. $\frac{1}{2}(V_1+V_2)$

2. $\frac{C_2{V}_1+C_1{V}_2}{C_1+C_2}$

3. $\frac{C_1{V}_1+C_2{V}_2}{C_1+C_2}$

4. $\frac{C_2{V}_1-C_1{V}_2}{C_1+C_2}$

Four capacitors are connected as shown in the figure. Their capacities are indicated in the figure. The effective capacitance between points x and y is (in μF) 

1. $\frac{5}{6}$

2. $\frac{7}{6}$

3. $\frac{8}{3}$

4. 2

What is the equivalent capacitance between A and B in the given figure (all are in farad) 

1. $\frac{13}{18}F$

2. $\frac{48}{13}F$

3. $\frac{1}{31}F$

4. $\frac{240}{71}F$

Two dielectric slabs of constant \(K_1\) and \(K_2\) have been filled in between the plates of a capacitor as shown below. What will be the capacitance of the capacitor? 

1. \(\frac{2\varepsilon_0A}{2}\left(K_1+K_2\right)\)
2. \(\frac{2\varepsilon_0A}{2}\frac{\left(K_1+K_2\right)}{K_1\times K_2}\)
3. \(\frac{2\varepsilon_0A}{d}\left(\frac{K_1+K_2}{K_1-K_2}\right)\)
4. \(\frac{2\varepsilon_0A}{d}\left(\frac{K_1\times K_2}{K_1+K_2}\right)\)

Three capacitors each of capacitance C and of breakdown voltage V are joined in series. The capacitance and breakdown voltage of the combination will be

1. $\frac{C}{3},\frac{V}{3}$

2. $3C,\frac{V}{3}$

3. $\frac{C}{3},3V$

4. $3C,3V$