If potential [in volts] in a region is expressed as V [x, y, z] = 6xy- y + 2yz, the electric field [in N/C] at point [1, 1, 0] is:

1. \(- \left(3 \hat{i} + 5 \hat{j} + 3 \hat{k}\right)\)

2. \(- \left(6 \hat{i} + 5 \hat{j} + 2 \hat{k}\right)\)

3. \(- \left(2 \hat{i} + 3 \hat{j} + \hat{k}\right)\)

4. \(- \left(6 \hat{i} + 9 \hat{j} + \hat{k}\right)\)

A parallel plate air capacitor has capacitance \(C,\) the distance of separation between plates is \(d\) and potential difference \(V\) is applied between the plates. The force of attraction between the plates of the parallel plate air capacitor is:
1. \(\frac{C^2V^2}{2d}\)
2. \(\frac{CV^2}{2d}\)
3. \(\frac{CV^2}{d}\)
4. \(\frac{C^2V^2}{2d^2}\)

A parallel plate air capacitor of capacitance \(C\) is connected to a cell of emf \(V\) and then disconnected from it. A dielectric slab of dielectric constant \(K,\) which can just fill the air gap of the capacitor is now inserted in it. Which of the following is incorrect?

Two thin dielectric slabs of dielectric constants K₁ and K₂ (K₁ < K₂) are inserted between plates of a parallel plate capacitor, as shown in the figure. The variation of electric field 'E' between the plates with distance 'd' as measured from plate P is correctly shown by: 

   

1.		2.
3.		4.

 

A conducting sphere of radius \(R\) is given a charge \(Q.\) The electric potential and the electric field at the centre of the sphere respectively are:

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. \(4\sqrt{35}~\text{N}\)

A, B and C are three points in a uniform electric field. The electric potential is: 

1. maximum at B
2. maximum at C
3. same at all the three points A, B and C
4. maximum at A

An electric dipole of moment \(p\) is placed in an electric field of intensity \(E.\) The dipole acquires a position such that the axis of the dipole makes an angle $\theta$ with the direction of the field. Assuming that the potential energy of the dipole to be zero when $\theta =90°$, the torque and the potential energy of the dipole will respectively be:
1. $pE\sin \theta ,-pE\cos \theta$
2. $pE\sin \theta ,-2pE\cos \theta$
3. $pE\sin \theta ,$ $2pE\cos \theta$
4. $pE\cos \theta ,-pE\sin \theta$

Four-point charges –Q, -q, 2q and 2Q are placed, one at each corner of the square. The relation between Q and q for which the potential at the center of the square is zero is:

1. Q = -q

2. Q = -2q

3. Q = q

4. Q = 2q

Two metallic spheres of radii \(1\) cm and \(3\) cm are given charges of \(-1\times 10^{-2}~C\) and \(5\times 10^{-2} ~C\), respectively. If these are connected by a conducting wire, then the final charge on the bigger sphere is:
1.  \(3\times 10^{-2}~ C\)
2.  \(4\times 10^{-2}~C\)
3.  \(1\times 10^{-2}~C\)
4.  \(2\times 10^{-2}~C\)