1.20 Relation between Field and Potential

We have discussed the electric field and potential. Electric potential is a scalar quantity but electric field is a vector quantity. How are these quantities related to each otehr?

                To derive the relation between electric field and potential let us consider two closely spaced equipotential surfaces A and B as shown in figure with potential values V and V +dow V , where Dow V is the change in V in the direction of electric field E. Let P be a point on the surface B. dow l is the perpendicular distance of the surface A from P. Imagine that a unit positive charge is moved along this perpendicular from the surface B to surface A against the electric field. The work done in this process is |E | dow l


           We thus arrive at two important conclusions concerning the relation between electric field and potential: (i) Electric field is in the direction in which the potential decreases steepest. (ii) Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point

1.19 Equipotential Surface

We have discussed the potential due to point charge, electric dipole, etc. Now let us go through the equipotential surface and its properties.
          An equipotential surface is a surface with a constant value of potential at all points on the surface.

Properties of Equipotential Surface
       This shows that V is a constant if r is constant. Thus, equipotential surfaces of a point charge are concentric spherical surfaces centred at the charge. Now the electric field lines for a  single charge q are radical lines starting from or ending at the charge, depending on whether q is positive or negative. Clearly, the electric field at every point is normal to the equipotential surface passing through that point. Theis is true in general: for any charge configuration, electric field at any point is normal to the electric field at that point. Why should it be normal?

             If the field were not normal to the equipotential surface, it would have non-zero component along the surface. To move a unit test charge against the direction of the component of the field, work would have to be done. But this is in contradiction to the definition of an equipotential surface: there is no potential difference between any two points on the surface and no work is required to move a test charge on the surface. The electric field must, therefore, be normal to the equipotential surface at every point.
              
              Equipotential surfaces offer an alternative visual picture in addition to the picture of electric field lines around a charge configuration. For a uniform electric field E, say, along the x- axis, the equipotential surfaces are planes normal to the x-axis, ie, planes parallel to the y-z plane.

3.20 Magnetisation and Magnetic Intensity

You have seen that materials like iron, steel etc are strongly attracted by external magnetic field, while wood, aluminium etc. do not show any attraction. What is the reason for these types of behaviour? If we observe a piece of magnetic material on an atomic scale we can see that the electrons orbiting around the nuclei and spinning about their own axes cause tiny currents. These current loops are so small that we can consider them as magnetic dipoles for practical purposes. Ordinarily they cancel each other because of the random orientation of atoms. But when a magnetic field is applied, a net alignment of these magnetic dipoles occurs, and the medium becomes magnetically polarized or magnetized. The net magnetic dipole moment developed per unit volume of a material is called intensity of magnetization or simply magnetization M (Its unit is A/m).

Different types of materials show different types of magnetization. Some materials acquire a magnetization parallel to the external applied magnetic field while some others opposite to it. There are substances which retain their magnetization ever after the external field has been removed (permanent magnets)

Consider a long solenoid of n turns per unit length and carrying current I. The magnetic field in the interior of the solenoid is
B0=mu0nI

The magnetic intensity H is a quantity related to currents in coils and conductors. In this case it is defined as
H= B0/mu0

H=nI

The magnetic intensity is a vector with dimensions of L-1A. Its unit is Am-1.

If we use another material such as iron as the core of the solenoid, keeping the current I constant, the magnetic field inside will be different from B0. This is because the core gets magnetized in the magnetic field produced by the current carrying coil, and this magnetized core in turn produces its own magnetic field. Let this magnetic material possess a dipole moment m. We define a relevant quantity called the magnetisation M which is equal to the magnetic moment per unit volume (V).

M=m/V

M is a vector with dimensions L-1A and unit Am-1. Thus M and H have the same units.
We can write the resultant field as B=B0+Bm
where Bm is the magnetic field produced due to the magnetization of the core. But Bm=mu0M.

Hence the total magnetic field in the material is B=mu0H+mu0M

B=mu0(H+M)

We have partitioned the contribution to the total magnetic field inside the solenoid into two parts: one, due to external factors such as the current in the solenoid, represented by H and the other is due to the specific nature of the magnetic material represented by M. The M can be influenced by external factors, which can be mathematically expressed as

M directly proportional to H or M=psi H

where psi is a dimensionless constant called the magnetic susceptibility. It is a measure of how a magnetic material responds to an external field. It is large and positive for materials which are called ferromagnetic. It is small and positive for materials which are called paramagnetic. It is small and negative for materials which are termed diamagnetic.

We obtain B=mu0(1+psi)H

B=mu0murH

where mur =(1+psi) is a dimensionless quantity called the relative permeability of the material inside the solenoid. For vaccum mur=1, psi=0

We can write mu0mur=mu, the absolute permeability of the medium

Therefore B=muH.

1.18 Electric Potential

Why do we define electric potential?        We have defined potential energy of a test charge q in terms of the work done on the charge q. This work is obviously proportional to q, (since the force at any point is qE, where E is the electric field at that point due to the given charge configuration). If we divide the work by the amount of charge q, the resulting quantity is independent of q. In other words, work done per unit test charge is characteristic of the electric field associated with the charge configuration. Thus work done per unit charge leads to the idea of electrostatic potential V due to a given charge configuration.

Wab/q = Vp
      
            This equation explain that the work done by an external force in bringing a unit positive charge from infinity to a point is equal to the electrostatic potential(V) at that point. Hence electro static potential at any point may be defined as the work done in bringing a unit positive charge from infinity to that point without any acceleration.


1.18 Potential due to a Point Charge
       The electric potential due to a point charge at a point depends on magnitude of charge, distance of the point and also on the surrounding medium. The equations shows that potential due to a point charge decreases with increase in distance and becomes zero at infinity

1.17 Electrostatic Potential Energy

        To get a clear idea about electrostatic potential energy, consider the field E due to a charge Q placed at the origin. Now, imagine that we bring a positive test chrge q from a point R to a point P against the repulsive force on it due to the charge Q.
         
             In this situation, work done by the external force(Fex) is the negative of the work done by the electric force, and gets fully stored in the form of potential energy of the charge q. Thus, work done by external forces in moving a charge q from R to P is

                   Wrp= Integral limit from R to P Fext.dr
                             (note here that this displacement is in an opposite sense to the electric force (E) and hence work done by electric field is negative,. This work done is against electrostatic repulsive force and gets stored as potential energy.

           At every point in electric field, a particle with charge q possesses a certain electrostatic potential energy, this work done increases its potential energy by an amount equal to potential energy difference between points R and P. Thus, potential energy difference.

       Therefore, we can define electric potential energy difference between two points as the work required to be done by an external force in moving (without accelerating) charge q from one point to another.

        We have discussed potential energy and potential energy difference, Which is more significant, potential energy or potential energy difference?
          
          Equation (2) defines potential energy difference in terms of the physically meaningful quantity work. The actual value of potential energy is not physically significant; it is only the difference of potential energy that is significant. To clarify this, let us add a constant value alfa to Up and Ur and find the difference

               This shows that, we can always add an arbitrary constant to potential energy at every point, since this will not change the potential energy difference:

                 The above argument gives a freedom in choosing the point where potential energy is zero without changing the value of potential energy difference. Potential energy at infinity is zero( We will learn this in next section). With this choice, if we take the point R at infinity, we get it.
       
             Since the point P is arbitrary, provides us with a definition of potential energy of a charge q at any point. Potential energy of charge q at a point ( in the presence of field due to any charge configuration) is the work done by the external force (equal and opposite to the electric force) in bringing the charge q from infinity to that point.

1.16 Electrostatic Potential

      In class XI, the notion of potential energy was introduced. When a body is taken from a potential energy was introduced. When a body is taken from a point to another against a force ( like spring force) the work done gets stored as potential energy of the body. When the external force is removed, the body retraces its path, gaining kinetic energy and losing an equal amount of potential enrgy. The sum of kinetic and potential enrgies is thus conserved. Forces of this kind are called conserved forces. Spring force and gravitational force are expamples of conservative forces.
             
               Coulomb force between two (stationary) charges, like the gravitational force, is also a conservative force. Thus, like the potential energy of a mass in a gravitational field, we can define electrostatic potential energy of a charge in an electrostatic field.
      
                  

3.19 The Earth's Magnetism

You have seen that a freely suspended or pivoted magnet comes to rest along the north-south direction. Experiments have shown that the earth behaves as a magnetized sphere. The strength of the earth's magnetic field varies from place to place on the earth's surface. Its value being of the order of 10 raised to -5 T.

The earths magnetic field was thought of as arising from a giant bar magnet approximately along the axis of rotation of the earth and deep in the interior. But this is certainly not correct. The magnetic field is now thought to arise due to electrical currents produced by convective motion of metallic fluids (consisting mostly of molten iron and nickel) in the outer core of the earth. This is known as dynamo effect.

The magnetic field lines of the earth resemble that of a (hypothetical) magnetic dipole located at the center of the earth. The axis of the dipoles does not coincide with the axis of rotation of the earth but is presently titled by approximately 11.3 degree with respect to the latter.

The location of the north magnetic pole is at a latitude of 79.74 degree N and a longitude of 71.8 degree W, a place somewhere in north Canada. The magnetic south pole is at 79.74 degree S, 108.22 degree E in the Antarctica. The magnetic poles are approximately 2000 km away from the geographic poles. The magnetic equator intersects the geographic equator at longitudes 6 degree W and 174 degree E, respectively.

The pole near the geographic north pole of the earth is called the north magnetic pole. Likewise, the pole near the geographic south pole is called the south magnetic pole. Since we know that unlike poles attract, the south pole of earth magnet is near the geographic north pole and vice versa.

Over small regions of space we can consider earth's magnetic field as uniform. So when a magnet is freely suspended in this uniform magnetic field it experiences a torque and aligns itself in the earth's field.

Elements of Earth's Magnetic Field

Consider a point P on the earth's surface. At this point, the longitude determines the north-south direction. The vertical plane containing the vertical axis and the longitude is called the geographic meridian.

At P, there also exists the earth's magnetic field Be. The magnetic meridian is the vertical plane containing the earth's magnetic field Be and the vertical axis.

The angle between the geographic and the magnetic meridian planes is called the magnetic derivation. Declination is greater at higher latitudes and smaller near the equator.

If a magnetic needle is perfectly balanced, about a horizontal axis is so that it can swing in a plane of the magnetic meridian, the needle would make an angle with the horizontal. Thus dip is the angle that the total magnetic field of the earth makes with the surface of the earth.