Electric Charges and Fields

Let there be a spherically symmetric charge distribution with charge density varying as   upto r = R, and ρ(r) = 0 for r > R, where r is the distance from the origin. The electric field at a distance r ( r < R) from the origin is given by

A long cylindrical shell carries positive surface charge  in the upper half and negative surface charge  in the lower half. The electric field lines around the cylinder will look like figure given in: (figures are schematic and not drawn to scale)

Two charges, each equal to q, are kept at x = −a and x = a on the x-axis. A particle of mass m and charge q_o =-q/2 is placed at the origin. If charge q_o is given a small displacement (y<< a) along the y-axis, the net force acting on the particle is proportional to

• y

• -y

• 1/y

• 1/y

The region between two concentric spheres of radii ‘a’ and ‘b’, respectively (see figure), has volume charge density ρ = A/r , where A is a constant and r is the distance from the centre. At the centre of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant is:

A charge Q is uniformly distributed over a long rod AB of length L as shown in the figure. The electric potential at the point O lying at a distance L from the end A is

In a uniformly charged sphere of total charge Q and radius R, the electric field E is plotted as a function of distance from the centre. The graph which would correspond to the above will be

From a uniform circular disc of radius R and mass 9M, a small disc of radius R/3 is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is:

• $\frac{37}{9}M{R}^2$

• 4MR²

• $\frac{40}{9}M{R}^2$

• 10 MR²

This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
An insulating solid sphere of radius R has a uniformly positive charge density ρ. As a result of this uniform charge distribution, there is a finite value of the electric potential at the centre of the sphere, at the
surface of the sphere and also at a point out side the sphere. The electric potential at infinity is zero.

Statement 1: When a charge q is taken from the centre to the surface of the sphere, its potential energy changes by qρ/3ε_o
Statement 2: The electric field at a distance r(r < R) from the centre of the sphere is  ρr/3ε_o

• Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for statement 1.

• Statement 1 is true, Statement 2 is false

• Statement 1 is false, Statement 2 is true

• Statement 1 is false, Statement 2 is true