Chapter 2 Electrostatic Potential and Capacitance

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 In this chapter, we discuss potential energy and electrical energy store. Potential energy can be calculated if the force field is conservative, and we know the electric force is conservative in nature.

Electrostatic Potential Energy

The work done to bring a charge from infinity to a given point in an electric field due to a fixed charge is called the potential energy.

W=kQq/r2

Electric Potential

The work done required in bringing a unit test charge from infinity to any point present in electric field through a fixed charge is called the electric potential at that point.

In terms of mathematics, if W is the required work done in bringing a test charge (q0) from infinity to point P (say), then potential will be

W=q0V

So the dimension of Electric potential will be [ML2T-3A-1], and its S.I. unit is volt (V).

Potential due to point charge

Consider a point charge Q placed at origin (O). Let P be any point where we have to determine potential and the distance between P and O is r. Now from the definition of potential we have to bring a unit test charge from infinity to point P.

Potential due to Electric Dipole


Let an electric dipole consists of two charges distance 2a. Its total charge is zero. Since the dipole moment vector p whose magnitude is q × 2a and which points in the direction from –q to q.

So the Electric potential

at its axis V=kp/r2

at its equatorial position is zero

st a general point having polar coordinates (r,θ) with respect to center of dipole is V= k (p cosθ)/r2

Potential due to system of charge particle

V= V1+V2+…….. + Vn

Equipotential Surface

An equipotential surface is a surface with a constant value of potential at all points on the surface.

POTENTIAL ENERGY OF A SYSTEM OF CHARGES

if there are n number of point charges q1, q2, q3… qn in system at separation rij between ith and jth, then potential energy of the system will be

U=Σi Σi<j[ {k qi qj}/rij ]

POTENTIAL ENERGY IN AN EXTERNAL FIELD

Potential energy of a single charge

Since as we know, we can determine potential energy of those particles whose source are specified. But here only one charge is present no source is there. So whenever ask these kinds of questions there are must have an external field always present.

So, the external electric field E and the corresponding external potential V may vary from point to point. By definition, V at a point P is the work done in bringing a unit positive charge from infinity to the point P. Thus, work done in bringing a charge q from infinity to the point P in the external field is qV. This work is stored in the form of potential energy of q. If the point P has position vector r relative to some origin, we can write:

Hence potential energy of q at r in an external field = qV(r)

Where V(r) is the external potential at the point r.

Potential energy of a system of two charges in an external field

The potential energy of the system is equal to the total work done in assembling the configuration

If q1 and q2 placed in an external field are the point charges at separation r12.

W= q1V(r) +q2V(r) + k q1 q2/r12

Potential energy of a dipole in an external field

Consider a dipole with charges q1 = +q and q2 = –q placed in a uniform electric field E.

In a uniform electric field, the torque τ experiences by dipole is,

τ= p x E

Suppose an external torque τext is applied in such a manner that it just neutralizes this torque and rotates it in the plane of paper from angle θ0 to angle θ1 at an infinitesimal angular speed and without angular acceleration. The amount of work done by the external torque will be given by

W= pE ( cosθ1 – cosθ2)

This work is stored as the potential energy of the system. We can then associate potential energy U(θ) with an inclination θ of the dipole.

If dipole is initially in stable equilibrium ( ) and final inclination is θ, then potential energy will be

W= pE ( 1 – cosθ)

ELECTROSTATICS OF CONDUCTORS

Conductors contain mobile charge carriers. In metallic conductors, these charge carriers are electrons. In a metal, the outer (valence) electrons part away from their atoms and are free to move. These electrons are free within the metal but not free to leave the metal. The free electrons form a kind of ‘gas’; they collide with each other and with the ions, and move randomly in different directions. In an external electric field, they drift against the direction of the field. The positive ions made up of the nuclei and the bound electrons remain held in their fixed positions.

Inside a conductor, electrostatic field is zero

Consider a conductor, neutral or charged. There may also be an external electrostatic field. In the static situation, when there is no current inside or on the surface of the conductor, the electric field is zero everywhere inside the conductor. This fact can be taken as the defining property of a conductor. A conductor has free electrons. As long as electric field is not zero, the free charge carriers would experience force and drift. In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside. Electrostatic field is zero inside a conductor.

At the surface of a charged conductor, electrostatic field must be normal to the surface at every point

If E were not normal to the surface, it would have some non-zero component along the surface. Free charges on the surface of the conductor would then experience force and move. In the static situation, therefore, E should have no tangential component. Thus electrostatic field at the surface of a charged conductor must be normal to the surface at every point. (For a conductor without any surface charge density, field is zero even at the surface.)

Electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface

Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface. That is, there is no potential difference between any two points inside or on the surface of the conductor. If the conductor is charged, electric field normal to the surface exists; this means potential will be different for the surface and a point just outside the surface.

In a system of conductors of arbitrary size, shape and charge configuration, each conductor is characterized by a constant value of potential, but this constant may differ from one conductor to the other.

DIELECTRICS AND POLARISATION

The insulators are often referred as dielectrics. Each dielectric is formed of atoms/molecules. In some dielectrics the positive and negative charge centers collide; such dielectric are said to be non-polar dielectrics while in some other dielectrics the center of positive and negative charges do not coincide, such dielectric have permanent electric dipole moment; such dielectrics are said to be polar dielectrics. The example of polar dielectric is water, and the example of non-polar dielectric is CO2.

When a dielectric is placed in an external field, the centers of positive and negative dipole separated (in non-polar dielectric) or get farther away (in polar dielectric), so that molecules of dielectric gain a permanent electric dipole moment this process is called polarization (P)and the dipole is polarised.

P= Χe E

CAPACITORS AND CAPACITANCE

Capacitor is system where we can store charge through creating a potential difference. Capacitor consists of two conductor placed at finite distance. And between the gaps there are some dielectrics (insulators) present. So, for storing charge the conductors must have some potential difference or they must have charged.

The capacitance of the conductor is the ratio between magnitude of charge at any plate and potential difference V across the plate i.e,

C=Q/V

The unit of capacitance is farad (F)

COMBINATIONS OF CAPACITORS

Parallel Combination

When capacitor is connected in parallel, then resultant capacitance will be

C = C1+ C2+ C3+…………………+ Cn

Series Combination

When capacitor is connected in series, then resultant capacitance will be

1/C = 1/C1+1/C2+……………….+1/Cn

Parallel Plate Capacitor

A parallel plate capacitor consists of two metallic plates separated by small distance and between the plates there is some kind of dielectric present. The capacitance of parallel plate is given by

C = Aε/d

where ε is the permittivity of the medium placed between two plates and A be the area of each plate and d the separation between them.

ENERGY STORED IN A CHARGED CAPACITOR

The potential energy stored in a charged capacitor in the following forms:

U= Q2/2C

U= 1/2 CV2

U= 1/2 QV

U = 1/2 ε0E2 x Ad

Where Ad is the volume of the capacitor. So energy stored in the capacitor per unit volume is

U = 1/2 ε0E2

The Van de Graaff Generator

In 1929 Robert J. Van de Graaff (1901–1967) used this principle to design and build an electrostatic generator. This type of generator is used extensively in nuclear physics research. This is a machine that can build up high voltages of the order of a few million volts. The resulting large electric fields are used to accelerate charged particles (electrons, protons, ions) to high energies needed for experiments to probe the small scale structure of matter.


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