Mathematics Electrostatic Potential and Capacitance

Definition Based Problems

Q 3211578429

Why is electrostatic potential constant throughout the volume of the conductor and has the same value (as inside) on its surface?
CBSE 12 Delhi

As we know that the electrostatic field inside the conductor is zero and on the surface, the field is normal to the surface at every point (Gauss's theorem).

No work is done in moving a small test charge, within the conductor and on its surface. We find there is no potential difference between the two points inside or on the surface, which implies the potential being constant throughout.
Q 3211678520

Explain the "For any charge configuration, equipotential surface through a point is normal to the electric field."
CBSE 12 Delhi 2014

If the electric field is not normal, it will have non-zero component along the surface. In that case, the work done in moving a charge on the equipotential surface will not be zero.
Q 3221678521

Define the geometrical shape of equipotential surfaces due to a single isolated charge?
CBSE 12 Delhi 2013

The geometrical shape is spherical.
Q 3241678523

There no work done in moving a charge from one point to another on an equipotential surface Why ?
CBSE 12 Foreign 2012

Potential difference between any two points of an equipotential surface is zero. So no work is done in moving a charge from one point to another.
Q 3271678526

Can two equipotential surfaces intersect each other?


Two equipotential surfaces cannot intersect. The direction of electric field is always perpendicular to the equipotential surface. If they intersect, there will be two directions of the electric field at the point of intersection which is not possible.
Q 3281678527

Why should electrostatic field be zero inside a conductor?
CBSE 12 Delhi 2012

In the static situation, there is no current inside, or on the surface of the conductor. Hence, electric field is zero everywhere inside the conductor.
Q 3201678528

Why does the electric field inside a dielectric decrease when it is placed in an external electric field?


An electric field is developed inside the dielectric due to induction in a direction opposite to the direction of external electric field. Thus, net electric field decreases.
Q 3251191024

Where does the energy of a capacitor reside?


Electrical energy resides in the space, within the plates.
Q 3251778624

Derivation for the electric potential at any point along the axial line of an electric dipole ?


Potential at `P` due to dipole is given by

`V_p = V_(PA)+V_(pB) = q/(4pi epsi_0) [ 1/(r-l) -1/(r+l)]`

` = 1/(4piepsi_0) (q. 2l)/((r^2-l^2))`

` = 1/(4pi epsi_0) p/((r^2-l^2))`
Q 3261778625

Draw a plot showing the variation of (i) electric field (E) and (ii) electric potential (V) with distance r due to a point charge Q.


Q 3271878726

Define is an electrostatic shielding? What is its practical importance?


Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded from the outside electric influence. This is known as an electrostatic shielding.

The effect can be made use of in protecting the sensitive instruments from the outside electrical influence.
Q 3281878727

Derive the expression for the capacitance of a parallel plate capacitor having plate area A and plate separation d.


Consider a capacitor with surface charge density cr on its plates. Suppose area of each plate is A and separation between the plates is d.

We know `Q = CV => C = Q/V`

Here `Q = sigmaA` .............(i)

`V = E_0 d = sigma/epsi_0 d \ \ \ \ \ \ \ \ ( because E_0 = sigma/epsi_0)` .............(ii)

From equations (i) and (ii), we get

`C_0 = ( epsi_0 A)/d`
Q 3211878729

Define the capacitor? Write its two uses.


A capacitor is an arrangement of primarily two conductors which is used for storing charge.
The capacitor is used for storing: (i) electrical charge, and (ii) electrical energy.
Q 3251078824

Explain 'dielectric constant' of a medium. Briefly explain why the capacitance of a parallel plate capacitor increases, on introducing a dielectric medium between the plates.


Dielectric constant of a medium is the ratio of the permittivity of the medium to that of vacuum.

An electric field is developed inside the dielectric due to induction in a direction opposite to the direction of external electric field. Thus, net electric field decreases.
Q 3221178921

Drive the expression for the energy stored in a parallel plate capacitor `C` having charges `+Q` and `-Q` on its plates.
CBSE 12 Delhi 2011

Let a capacitor of capacity `C` be charged to a potential `V` by a charge `Q`. Work has to be done in charging a capacitor. This work is stored in the capacitor in the form of potential energy. It exists in the electric field between the plates of the capacitor.

Let `q` be the charge on the capacitor at any stage and `V` the potential, then

`V = q/C`

Then small amount of work done `dw` in giving a further small charge `dq` to the capacitor is

`dw = V dq = q/C dq`

`therefore` Work done in charging the capacitor to a charge `Q` is

`W = int_0^Q q/C dq = 1/2 [q^2/C]_0^Q = 1/(2C) [ Q^2-0] = 1/2 Q^2/C`

Also, `Q = CV`

`therefore W = 1/2 ( C^2 V^2)/C = 1/2 CV^2`
Q 3211180929

Explain the working principle of a parallel plate capacitor. On what factors, the capacitance of a parallel plate capacitor depends?


Principle: When an uncharged, earthed conductor is brought near to a charged conductor, then the potential of later decreases and its charge holding capacity increases.

The capacitance depends on:
(i) Geometrical configuration (shape, size and separation) of the system of two conductors.
(ii) Nature of the dielectric separating two conductors.
Q 3251178924

Derive an expression for the potential energy of an electric dipole of dipole moment `vecp` in an electric field `vecE`


Torque acting on the dipole, `tau = p E sin theta`

It tends to rotate the dipole in clockwise direction. To rotate the dipole anticlockwise
work has to be done on the dipole.

`W = int_(theta_1)^(theta_2) tau d theta = int_(theta_1)^(theta_2) p E sin theta d theta = -p E ( costheta_2- costheta_2)`
Q 3201178928

Why is the dielectric constant of conductors taken as `oo`?


When the conductors are placed in the external field, then the induced electric field is equal and opposite to the external field `E_0`

`therefore ` Net field, `E_("Net") = E_0 - E_("in") = E_0 - E_0 = 0`

`because` Dielectric constant, `K = E_0/E_("Net") = E_0/0 = oo`
Q 3251180024

Drive the expression for the potential energy of a system of two point charges `q_1` and `q_2` brought from infinity to the points `vecr_1` and `vecr_2` respectively in the presence of external electric field `vecE`
CBSE 12 Delhi 2010

Work done in bringing the charge `q_1` from infinity to `vecr_1`

against the external electric field `W_1 = q_1 V ( r_1)`

Work done in bringing the charge `q_2` from infinity to `vecr_2 , W_2 = q_2 V ( r_2)`

Work done on `q_2` against the field due `q_1 , W_3 =( q_1 q_2)/(4pi epsi_0 r_(12))`

As work done is stored in the form of potential energy, therefore

Potential energy of the system ` = W_1+W_2+W_3 = U_1 + U_2+U_3`

Potential energy of the system `= q_1 V (r_1) + q_2V (r_2) + ( q_1 q_2)/(4pi epsi_0 r_(12))`
Q 3211280120

Define the polarization of charge? With the help of a diagram show why the electric field between the plates of capacitor reduces on introducing a dielectric slab. Define the dielectric constant on the basis of these fields.
CBSE 12 Delhi

The induced dipole moment developed per unit volume in a dielectric slab on placing it inside the electric field is called polarisation.

Let `vecE_0` be the uniform external electric field. When a dielectric slab is placed in uniform electric field, then the molecules get polarised, due to which `- sigma_p` (charge density due to polarisation) will appear near the positive plate and `+ sigma_p` will appear in the dielectric near the negative plate.

Therefore, due to polarization of molecules, electric field will appear will appear in the opposite

direction, and the net electric field inside the dielectric will be

`vecE= vecE_0 - vecE_p < vecE_0`

to the electric field in medium.