Electric Charges and Fields

Consider an electric dipole comprising two charges +q and -q each with mass m, separated by a fixed distance d and initially at rest with its dipole moment pointing along i. A uniform electric field E j is turned on at time t=0 and it is turned off at t=tf, when the dipole moment makes an angle f with i. Neglecting any sources of energy loss, correct option(s) is/are:

1. The center of mass of the dipole is deflected towards j in the presence of the field.
2. If the magnitude of the final angular velocity f= 2qEmd, then f= 6.
3. If f= /3, then the change in kinetic energy of the dipole is given by 2 3 qEd.
4. For f= /4, the dipole rotates around its center of mass with a constant angular velocity after t>tf.

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude p, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that they are at a distance 2r apart along the x direction. The midpoint of the line joining the two dipoles is X. The possible resultant electric fields E at X are given in List-II. Choose the option that describes the correct match between the entries in List-I to those in List-II.

1. P → 3, Q → 1, R → 2, S → 4
2. P → 4, Q → 5, R → 3, S → 1
3. P → 2, Q → 1, R → 4, S → 5
4. P → 2, Q → 1, R → 3, S → 5

Two co-axial conducting cylinders of same length with radii 2R and 2R are kept, as shown in Fig. 1. The charge on the inner cylinder is Q and the outer cylinder is grounded. The annular region between the cylinders is filled with a material of dielectric constant =5. Consider an imaginary plane of the same length at a distance R from the common axis of the cylinders. This plane is parallel to the axis of the cylinders. The cross-sectional view of this arrangement is shown in Fig. 2. Ignoring edge effects, the flux of the electric field through the plane is ( 0 is the permittivity of free space): [figure]

1. Q30 0
2. Q15 0
3. Q60 0
4. Q120 0

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of 0. The separation between any two consecutive sheets is 1 . The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If 0=9 C/m2, then which of the following statements is/are correct: (Take permittivity of free space 0=9 10-12 F/m) [figure]

1. In region 4 of the configuration I, the magnitude of the electric field is zero.
2. In region 3 of the configuration II, the magnitude of the electric field is 0 0.
3. Potential difference between the first and the last sheets of the configuration I is 5 V.
4. Potential difference between the first and the last sheets of the configuration II is zero.

A small electric dipole p0, having a moment of inertia I about its center, is kept at a distance r from the center of a spherical shell of radius R. The surface charge density is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle as shown in the figure. While staying at a distance r, the dipole is free to rotate about its center. [figure] If released from rest, then which of the following statement(s) is(are) correct? [ 0 is the permittivity of free space.]

1. The dipole will undergo small oscillations at any finite value of r.
2. The dipole will undergo small oscillations at any finite value of r > R.
3. The dipole will undergo small oscillations with an angular frequency of 2 p0 0 I at r = 2R.
4. The dipole will undergo small oscillations with an angular frequency of p0100 0 I at r = 10R.

A charge is kept at the central point P of a cylindrical region. The two edges subtend a half-angle at P, as shown in the figure. When = 30 , then the electric flux through the curved surface of the cylinder is . If = 60 , then the electric flux through the curved surface becomes / n, where the value of n is . [figure]