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.

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.

1.12 Electric Flux

       Consider the flow of a liquid with a velocity v through a pipe of area of cross section dS. The rate of flow of liquid (V dS) represents the flux of liquid flowing across the plane. In the case of electric field, we define an alnalogous quantity and call it the electric flux.
          Flux of liquid gives rate of flow of liquid, then what does electric flux represent?
       We have discussed that the relative density of the field lines at different points indicates the relative strength of electric field at those points. Here the mathematical quantity E delta S (electric flux) gives a measure of lines passing through the area delta S.
       On which factors do E delta S (electric flux) depend?
                      Consider a small planar element of area delta S placed normal to vector.  The number of field lines crossing the area element will be smaller. The projection of the area element normal to E is deltaS cos (theta). Thus, the number of field lines crossing delta S is proportional to EdeltaS cos(theta)
                        When theta=90 degree, field lines will be parallel to plane and will not cross it at all. From the above discussion we can understand that the electric flux depend on electric field E, area A and the orientation of area with electric field.

                       From the above discussion we can understand that the electric flux depend on electric field E, area A and the orientation of area with electric field.
                      
                        If vector E is the electric field and delta vector S is an area, the electric flux may be defined as vector E. delta vector S(dot product of vector E and delta S). Area is treated as a vector. Its direction is along the normal to the plane of area.
                   
                       If this small area is a part of a large surface, then electric flux over the surface, then electric flux over the surface.


                    

3.15 The Bar Magnet

What happens when a magnet is brought nearer to an iron rod floated on water or suspended by a string from its center? Try to perform this activity and note down your observation.

The rod tends to line up in a north-south direction. The tip which points to the geographic north is called north pole, and the tip which points to the geographic south is called south pole.

When two magnets are brought nearer to each other, like poles repel each other and opposite poles attract each other. An object that contains iron but is not itself magnetized (that is, it shows no tendency to point north or south) is attracted by either pole of a permanent magnet. This is the attraction that acts between a magnet and the unmagnetized steel door of a refrigerator.

When iron filings are sprinkled on a sheet of glass placed over a short bar magnet, a pattern is obtained. A similar pattern of iron filings is observed around a current carrying solenoid.

3.14 Force Acting Between Two Parallel Current Carrying Conductors

When two parallel current carrying conductors are close enough they exert force on each other. This can be realized by doing the following activity

Group Activity

Arrange two thin aluminium foils (10 cm x 10 cm) on two pencils with the help of crocodile pins. Connect a dc source (9V) to the foils such that the direction of current is same in both the foils. Note down your observation. Now reconnect the dc source such that the direction of current in the foils are in the opposite direction. Note down the new observation also. Now try to find the reason for your observation in a group discussion.

Through the above activity we can realize that the conductors attract each other when the current in them are parallel and repels when the current is antiparallel. In the above experiment each conductor is situated in the magnetic field due to the other. This is the reason of the force of interaction between them. Our next aim is to obtain a general expression of force acting between two parallel current carrying conductors.

Consider two very long conductors carrying currents I1 and I2 arranged parallel at a distance r apart. The magnetic field due to the first conductor at a distance r away from it is

B=muzero/4pix 2I1/r

The second conductor carrying current is placed in this magnetic field. Hence the force experienced by a length l of it is BI2l.

The force experienced per unit length is given by F=BI2=muzero/4pix 2I1I2/r

In SI system the unit of electric current intensity is defined using the force experienced by the parallel conductors carrying current. The definition is as follows:

One ampere is defined as that current which when flowing through each of two parallel conductors of negligible cross section and infinite length placed 1m apart in free space would impart a force of 2x10-7 N/m on each other.

3.13 Conversion of Galvanometer into Voltmeter

Voltmeter is a device used to measure pd between two points in a circuit. Voltmeter is connected in parallel between the points whose pd is to be measured. The introduction of voltmeter should not alter the pd or current in the circuit. Ideally voltmeter should not draw current from the  circuit. Hence an ideal voltmeter should have infinite resistance.

A galvanometer can be converted into a voltmeter by connecting a high resistance in series with it. Let maximum safe current through a galvanometer of resistance G be Ig and the voltage to be measured is V. Then the series resistance required is calculated as follows:

pd across voltmeter = max. Voltage

ie Ig(G+R)=V

or R= V/Ig -G

Thus a galvanometer of resistance G and safe current Ig can be converted into a voltmeter to read a maximum voltage of V by connecting a series resistance of value R=V/Ig -G

3.12 Conversion of Galvanometer into Ammeter

Ammeter is a device used to measure current in a circuit. It is connected in series with the circuit. The introduction of ammeter should not alter the pd or current in the circuit. Ideally there should be no potential drop across the ammeter. Hence an ideal ammeter has zero resistance.

A galvanometer can be converted into an ammeter by connecting a low resistance in parallel with it.

Let the maximum safe current through a galvanometer of resistance G is Ig and current I is flowing through the circuit. Then the purpose of shunt is to provide a short cut path for I-Ig current without affecting the galvanometer.

However the pd across the galvanometer and the shunt will be one and the same.

pd across shunt=pd across galvanometer.

ie, (I-Ig)S=IgG

S=Ig/(I-Ig)G

Thus a galvanometer of resistance G and maximum safe current Ig can be converted into an ammeter to read a maximum current of I by connecting a shunt of values =Ig/(I-Ig)G across it..

3.11 Moving Coil Galvanometer

Moving Coil Galvanometer (MCG) is a device used to detect small current. With slight modifications it can be used to measure current or voltage. MCG was first devised by Kelvin and later modified by D' Arsonval.

The working principle of MCG is that a freely suspended current carrying coil in a uniform magnetic field experiences a torque proportional to the current through the coil. MCG consist of a multi turn coil free to rotate about a vertical axis in a uniform radial magnetic field. There is a cylindrical soft iron core to increase the sensitivity of MCG.

Working

When a current flows through the coil the maximum torque experienced is T =NIAB and as a result the coil gets rotated. The suspension system of the coil supplies a restoring couple proportional to the angle of twist theta. If c is the torsion constant of the suspension system, then T=ctheta.

At equilibrium the external magnetic torque is balanced by the restoring torque.

Tmagnetic=Trestoring

substituting the value of torque

NABI= ctheta

I=(c/NAB)theta.

The bracket in the above expression is a constant for a given arrangement and is known as galvanometer constant k.

Therefore I=k theta.

By measuring theta for a small known current k can be determined. If k is determined once, then any unknown current can be determined by measuring the angle theta alone.

Rectangular Coil in a
Magnetic Field

Conversion of  Galvanometer
into Ammeter