Electrostatic Potential and Capacitance

A parallel plate capacitor is made up of stair like structure with a plate area A of each stair and that is connected with a wire of length b, as shown in the figure. The capacitance of the arrangement is $\frac{x}{15}\frac{{\in }_{0}A}{b},$ <!-- [if gte mso 9]><xml> </xml><![endif]--> the value of x is……………

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

$({\mathit{E}}_2$ - ${\mathit{E}}_1).\widehat{\mathit{n}}=\frac{\mathit{\sigma }}{{\mathit{\epsilon }}_0}$

Where $\widehat{\mathit{n}}$ is a unit vector normal to the surface at a point and s is the surface charge density at that point. (The direction of $\widehat{\mathit{n}}$ is from side 1 to side 2.) Hence, show that just outside a conductor, the electric field is $\mathit{\sigma }\widehat{\mathit{n}}$ / ${\mathit{\epsilon }}_0$ .

(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.]

2.31 (a) Two large conducting spheres carrying charges ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ are brought close to each other. Is the magnitude of electrostatic force between them exactly given by ${\mathit{Q}}_1{\mathit{Q}}_2$ /4 $\mathit{\pi }{\mathit{\epsilon }}_0{\mathit{r}}^2$ , where r is the distance between their centres?

(b) If Coulomb’s law involved 1/ ${\mathit{r}}^3$ dependence (instead of 1/ ${\mathit{r}}^2$ ), would Gauss’s law be still true ?

(c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point?

(d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical?

(e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there?

(f) What meaning would you give to the capacitance of a single conductor?

(g) Guess a possible reason why water has a much greater dielectric constant (= 80) than say, mica (= 6).

A parallel plate capacitor is connected to a battery as shown in figure. Consider two situations.

A. Key K is kept closed and plates of capacitors are moved apart using insulating handle.

B. Key K is opened and plates of capacitors are moved apart using insulating handle.

Choose the correct option(s).

(a) In A, Q remains the same but G changes

(b) In B, V remains the same but C changes

(c) In A, V remains the same hence Q changes

(d) In B ,Q remains the same hence V changes

If a conductor has a potential V≠0 and there are no charges anywhere else outside, then 

(a) There must be charges on the surface or inside itself

(b) There cannot be any charge in the body of the conductor

(c) There must be charges only on the surface

(d) There must be charges inside the surface

In the circuit shown in figure initially key K₁is closed and key K₂ is open. Then K₁ is opened and K₂ is closed (order is important). [Take Q’₁ and Q’₂ as charges on C₁ and C₂ and V₁ and V₂ as voltage respectively.] Then, E

(a) Charge on C, gets redistributed such that V₁ = V₂

(b) Charge on C₁ gets redistributed such that Q’₁ = Q’₂

(c) Charge on C₁ gets redistributed such that C₁V₁ + C₂V₂ = C₁E

(d) Charge on C₁ gets redistributed such that Q’₁ + Q’₂=Q

In a region of constant potential

(a) The electric field is uniform

(b) The electric field is zero

(c) There can be no charge inside the region

(d) The electric field shall necessarily change if a charge is placed outside the region

The work done to move a charge along an equipotential from A to B

Equipotential surfaces

(a) are closer in regions of large electric fields compared to regions of lower electric fields

(b) will be more crowded near sharp edges of a conductor

(c) will be more crowded near regions of large charge densities

(d) will always be equally spaced

Consider a uniform electric field in the z -direction. The potential is a constant

(a) in all space                             

(b) for any x for a given z

(c) for any y for a given z             

(d) on the x-y plane for a given z