Physics Class 12 NCERT Solutions: Chapter 2 Electrostatic Potential and Capacitance Part 7

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Surface of density

Surface of Density

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Q: 15. A spherical conducting shell of inner radius and outer radius has a charge

(A) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?

(B) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.

Answer:

(A) Charge placed at the centre of a shell is . Hence, a charge of magnitude will be induced to the inner surface of the shell. Therefore, total charge on the inner surface of the shell is .

Surface charge density at the inner surface of the shell is given by the relation,

A charge of is induced on the outer surface of the shell. A charge of magnitude Q is placed on the outer surface of the shell. Therefore, total charge on the outer surface of the shell is. Surface charge density at the outer surface of the shell,

(B)Yes

The electric field intensity inside a cavity is zero, even if the shell is not spherical and has any irregular shape. Take a closed loop such that a part of it is inside the cavity along a field line while the rest is inside the conductor. Net work done by the field in carrying a test charge over a closed loop is zero because the field inside the conductor is zero. Hence, electric field is zero, whatever is the shape.

Q: 16. (A)Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by

Where, is a unit vector normal to the surface at a point and iƒ is the surface charge density at that point. (The direction of is from side 1 to side 2.) Hence, show that just outside a conductor, the electric field is

(B) Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another. [Hint: For (A), use Gauss’s law. For, (B) use the fact that work done by electrostatic field on a closed loop is zero.]

Answer:

(A) Electric field on one side of a charged body is and electric field on the other side of the same body is E2. If infinite plane charged body has a uniform thickness, then electric field due to one surface of the charged body is given by,

Where,

Unit vector normal to the surface at a point

Surface charge density at that point

Electric field due to the other surface of the charged body,

Electric field at any point due to the two surfaces,

Since inside a closed conductor,

Therefore, the electric field just outside the conductor is.

(B) When a charged particle is moved from one point to the other on a closed loop, the work done by the electrostatic field is zero. Hence, the tangential component of electrostatic field is continuous from one side of a charged surface to the other.

Q: 17. A long charged cylinder of linear charged density is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?

Answer:

Charge density of the long charged cylinder of length and radius is . Another cylinder of same length surrounds the pervious cylinder. The radius of this cylinder is R.

Let be the electric field produced in the space between the two cylinders.

Electric flux through the Gaussian surface is given by Gauss’s theorem as,

Where, Distance of a point from the common axis of the cylinders Let q be the total charge on the cylinder.

It can be written as

Where,

Charge on the inner sphere of the outer cylinder

Permittivity of free space

Therefore, the electric field in the space between the two cylinders is .