1.15 Applications of Gauss's law

(a) Field due to an infinitely long straight uniformly charged wire
                To calculate the electric field, consider an infinitely long straight line of charge, having linear charge density lamda. Let P be a point at a perpendicular distance 'r' from the line of charge. To find the field at P, imagine a cylinder of radius 'r' having length 'l' with its axis as line of charge. Here, the cylinder is considered as Gaussian surface. P is a point on the GAussian surface.

                 The electric flux through the two circular faces is zero since electric field is normal to the area vector. Electric flux through the curved surface of cylinder.

                    By Gauss's theorem, total flux is equal to 1/epsilon zero times the net charge enclosed by the cylindrical surface. The direction of this field is normal to the curved surface passing normally to it.

(b) Electric Field Due to Infinite Plane Sheet of Charge
                   
                     Consider a plane  sheet of charge of surface density sigma. Gaussian surface which is a pillbox of area of cross section A as shown. Electric flux through the curved surface of the pill box will be zero, because electric field and area vector are normal to each other.

                       If sigma is positive E is directed normally out of the plate and if sigma is negative E is directed normally in to the plate.

1.14 Continuous Charge Distribution

       We have so far dealt with charge configurations involving discrete charges q1,12,....,1n. One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus. For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions.
       
         Consider a case that grains are spread uniformly over the floor, just one layer so that the floor is not at all seen. Now, what is the total number of grains? It is a laborious task to count one by one. So what can we do? Let us consider a unit area and count the number of grains in that area. Multiply this number by the total area and we can get the total number of grains. The number of grains per unit area can be called as surface grain density.
        
           Similarly, when charges are distributed over a finite space, it is useful to consider the density of charge. It is used in three different ways.

            i. Linear Charge Density
 (1) It is the charge per unit length. If Q charges distributed uniformly over a length 'l' then

                                lamda = Q/l

            ii. Surface Charge Density:
          It is the charge per unit area. If 'Q' charge is distributed uniformly in area 'A', then surface charge density sigma = Q/A

             iii. Volume Charge Density
           It is the charge per unit volume. If 'Q'  charge is distributed uniformly in a volume V, then volume charge density row = Q/V

2.11 Applications Of Thermo Electric Effects

Measurement of temperature and detection of radiation are the two important applications of Seebeck effect.

Thermoelectric Thermometer
To measure temperature , we use  thermocouple consisting of copper and constantan with one junction in contact with the region whose temperature is to be measured and the other junction in a constant temperature bath.
One end of the thermocouple is kept at a standard temperature , say ice at triple point and the other end of the couple kept at the region of known temperature . An electronic Voltmeter is connected in series with the arrangement to measure thermo emf. The thermocouple is the preferred way of measuring temperature , because of its accuracy , and convenience . Since the junction is small it absorbs very little heat, and quickly attain the temperature being measured. Copper-gold iron alloy thermocouple is used to measure 1K to 50K.
Copper-constantan thermocouple is used to measure 50K to 400K. Platinum-Rhodium Platinum alloy thermocouple is used to measure 1500K to 2000K.

Thermopile
Thermopile consists of large no of antimony bismuth thermo couples connected in series. One set of the junctions are kept at constant temperature. A galvanometer included in series is used to measure the  thermoelectric voltage. When thermal radiation falls on balckened junction its temperature rrises and thermo-emf proportional to the intensity of radiation is developed. Thus by noting the amount of deflection or from the thermo-emf developed the intensity of radiation can be obtained.

3.18 Magnetism and Gauss's Law

In chapter 1, we studied Gauss's law for electrostatics. For a closed surface the number of lines leaving the surface is equal to the number of lines entering it. This is consistent with the fact that no net charge is enclosed by the surface. However, in the same one, for the closer surface, there is a net outward flux, since it does include a net (positive) charge.

The situation is radically different for magnetic fields which are continuous and form closed loops. Both cases visually demonstrate that the number of lines of force leaving the surface is balanced by the number of lines entering it. The net magnetic flux is zero for both the surfaces. This is true for any closed surface.

Consider a small vector area element delta S of a closed surface S. The magnetic flux through delta S is defined as delta phi B = B.x delta S. where B is the field at delta S. We divide S into many small area elements and calculate the individual flux through each. Then the net flux phi B is,

phi B= sigma phi B= sigma B.delta S=0

where all stands for all area elements delta S.

Thus Gauss law for magnetism is:

The net magnetic flux through any closed surface is zero. The difference between the Gauss's law of magnetism and that for electrostatics is a reflection of the fact that isolated magnetic poles (also called monopoles) are not known to exist.

1.13 Gauss's Law

Gauss's law in electrostatics explains the relation between flux related to a surface and total charge enclosed by the surface. To derive the relation, consider a charge 'q' kept at the centre of a spherical surface of radius 'r'.

a) The magnitude of electric field on the surface.
b) We can consider that the spherical surface is made up of a number of small plane area each equal to 'dS'. The angle between the direction of electric field vector and area vector is zero.
c) The electric flux through such an area element ds

d) The electric flux through the entire spherical surface.


                              ie.. Total flux over a closed surface in free space is(one/epsilone) times the total charge enclosed by the surface. This is called Gauss's theorem.

                                        If there are several charges q1,q2,q3,. inside the closed surface, each will contribute to the total electric flux. Total electric flux is the summation * the 1/epsilon

         Where epsilon q is the algebraic sum of the charges within the surfaces.

Note:
 1) Since the electric lines are emerging from the charge 'q' the electric flux will not change even if the spherical surface of arbitrary shape. So an appropriate Gaussian surface is selected based on the symmetry of the problem under consideration. The examples illustrated in the coming section demonstrates the choice of Gaussian surface.

2) Now, as far as the charge is anywhere within the closed surface, the electric flux will not change. So we can say that the total flux through a closed surface is 1/epsilonzero times charge enclosed by the surface.

3) But if we keep an electric dipole within the closed surface of arbitrary shape, net flux from the closed surface is zero. The is because the electric lines emerging from positive charge ends on negative charge. Hence no net outward or inward flux, i.e, when the net charge inside the  surface is zero, the net flux through a closed surface is zero. If the charge enclosed is positive, the flux is outward and if it is negative, flux is inward.

3.17 The Dipole in a Uniform Magnetic Field

A magnetic dipole placed in the direction of field experiences no resultant force or torque, since the force on both the north and south poles are equal and opposite force along same line. But if the dipole is placed inclined to the field then both the north and south poles experiences equal and opposite force with different line of action. Then the magnetic dipole experience a resultant torque.

If m is the moment of magnet and B is the magnetic field intensity, then torque is given by

T=mxB

ie., T=mBsin theta..

3.16 The Magnetic Field Lines

The magnetic field lines (magnetic lines of force) are a visual and intuitive realization of the unseen magnetic field. The field lines of magnetic field lines for both the bar magnet and  the current carrying conductor and an electric dipole are very similar. The curves are closed Gaussian surfaces. Unlike electric lines of force, the magnetic field lines do not indicate the direction of the force on a moving charge.

Properties of magnetic field lines

1. The magnetic field lines form closed loops
2. The tangent to the field line at a given point represents the direction of the net magnetic field B at that point.
3. The number of field lines crossing per unit normal area is a measure of the magnitude of the magnetic field B.

The magnetic field lines do not intersect. This is because, if they intersect, it would mean that at the point of intersection, the field can have two directions, which is impossible.

One can plot the magnetic field lines in a variety of ways. One way is to place a small magnetic compass needle at various positions and note its orientation. This gives us an idea of the magnetic field direction at various points in space.

The bar magnet as an equivalent solenoid

What is the equation for magnetic dipole moment m associated with a current loop?

m=NIA, where N is the number of turns in loop, I the current and A the area vector.

The resemblance of magnetic field lines for a bar magnet and a solenoid suggest that a bar magnet may be thought of as a large number of circulating currents in analogy with a solenoid. Cutting a bar magnet in half is like cutting a solenoid. We get two smaller solenoids with weaker magnetic properties. The field lines remain continuous, emerging from one face of the solenoid and entering into the other face. One can test this analogy by moving a small needle in the neighbourhood of a bar magnet and a current carrying solenoid and noting that the deflections of the needle are similar in both cases.

It can be shown that the magnitude of the axial field of a solenoid at a distance x from its center,

B=mu0/4pi x 2m/x3

This is also the far axial magnetic field of a bar magnet which one may obtain experimentally. Thus, a bar magnet and a solenoid produce magnetic field. The magnetic moment of  a bar magnet is thus equal to the magnetic moment of an equivalent solenoid that produces the same magnetic field.