Electricity and magnetism pdf free download

Consider a number of concentric con- ducting spheres. Let the radius of the innermost be a and let the inner and outer radii of the spheres next in order be 6, V; c, c'; etc. If the charges on the spheres are given we can find the potentials, and conversely.

Let the charges be e, e', e", Again the spheres divide space into regions numbered 1, 2, 3, Condenser formed of coaxial cylinders. Use of Laplace's equation. There are many simple applications of this fact, especially where symmetry enables us to reduce the number of independent variables in the equation. We will take as illustrations the three types of condensers considered in , and Sets of condensers.

A set of n condensers is said to be arranged in series when the plate of lower potential of the first is connected with the plate of higher potential of the second, the plate of lower potential of the second to the plate of higher potential of the third and so on as in the figure. Let Clt C2, Cn be the capacities of the condensers, and let a charge e be given to the outer plate of the first. It follows that if n condensers of the same capacity G are joined in parallel the capacity of the compound condenser is nC.

Energy of a charged conductor. Consider a conductor alone in the field so that the potential is due only to the charge on the conductor. Energy of a charged condenser. Let a condenser have one of its plates earth-connected. Application to the parallel plate condenser. The explanation of this lies in the fact that to maintain the plates at a constant difference of potential they would have to be connected to the terminals of a battery, i. A general proof of the theorem of which this is a special case will be given in the next chapter.

Approximate expression for the capacity of a con- denser. Let a condenser be formed of two nearly parallel conducting surfaces. Electrostatic units for fields in air. The absolute electrostatic unit of charge in the O. The practical units are multiples or sub-multiples of the absolute units.

I, The reader should notice that the arguments by which we deduce the dimensions of these physical quantities in terms of mass, space and time will need qualification when we con- sider fields in other media than air; also, that there is another system of units—the electromagnetic—which starts from a different basis and differs from the electrostatic system.

Examples, i Two condensers of capacities microfarad and microfarad respectively are charged to voltages 1 and 10 respec- tively. Shew that if they are then connected in parallel there is a loss of energy amounting to ergs. But their total charge is abs. And the energy of the compound condenser is half the charge multiplied by the potential difference, i.

It is assumed that to the approximation required he lines offorce between the plates are nearly straight and thefieldoutside is negligible. We shall prove that the equilibrium is unstable by shewing that when the movable plate is turned through a small angle 6 it is acted on by a couple tending to increase the displacement.

Shew that the 'repulsion between them reduces to a single force through the centre of the sphere and that its component in any direction is equal to Qi8l 8nal , where S is the area of the curve formed by the projection on the plane normal to that direction of the curve dividing the two parts. Hence the element makes a contribution 2ttald8 to the required force. It is easy to see that if we take the direction of the required component force to be at right angles to the plane of the paper, the projection of the boundary will have a node.

But the portion ONKH of the sphere will contribute a force acting upwards through the paper, and the portion KNOML will contribute a force acting downwards through the paper, so that the required component of the resultant force perpendicular to the plane of the paper will correspond to the difference of the pro- jections of these areas, which will be the difference of the two loops into which S is divided through having a double point, and it is in this way that we must interpret S when its boundary has a double point.

Find the charge x which the electroscope receives. The two spheres are now connected to each other without affecting the divergence of the leaves of the electroscope, and the final charge on the electroscope is thus either x or —x. Explain why the disc commences to rotate. After a time the rate of rotation of the disc steadies down to a constant value, the moments of the electrical and frictional forces balancing.

There is then a re- pulsion between the charge at A and the charge at the opposing point of the disc, and as sparking is an irregular process this latter charge will not be uniformly distributed and the equilibrium will be unstable so that the disc begins to rotate. Calculate the capacity of a condenser formed by two circular plates of tinfoil mounted on glass, taking the diameter of eaoh plate to be 40 cm.

Express the result in terms of a microfarad. One plate of a parallel plate condenser is a circle of radius 10 cm. The distance between the plates is 5 mm. Prove that when the force of attraction between the plates is 5 grams weight the difference of potential between the plates is approxi- mately volts. The trap-door of an electrometer is a circle of 6 cm. Calculate the potential difference in volts.

The trap-door of a guard ring electrometer is circular, 6 cm. What difference of potential in electrostatic units between the plates of a guard ring electrometer is required to produce an attraction on the trap-door of 5 dynes per square centimetre, when the distance between the plates is centimetre?

What is the potential difference in volts? Three parallel plates A, B,G, each 10 cm. B is between A and C, A and B being separated by 1 mm. Find the magnitude and direction of the force required to prevent B moving. Shew that if the distance between the plates of a parallel plate condenser be halved the electric energy is halved or doubled according as the charges or the potentials are kept constant during the change.

Explain this on physical grounds. Neglecting edge effects, find the capacity of a condenser in air consisting of two sets of parallel square plates of side a each at distance h from its neighbours, and connected together alternately as in the diagram.

If the outer set are fixed in position and the inner set are free to move all together to the right or to the left, determine, from a consideration of the mechanical strains on the plates, whether the equilibrium of the central position is stable or unstable.

A conductor is charged from an electrophorus by repeated contacts with a plate which after each contact is re-charged with a quantity E of electricity from the electrophorus.

Prove that, if e is the charge of the conductor after the first operation, the ultimate charge is Eej E-e. They are initially uncharged, and A is permanently connected to earth potential zero. The two con- densers are connected in series AB Fig. Shew that there is a loss of energy equal to What becomes of this energy?

Two condensers A and B both consist of a pair of parallel circular plates of radius r. The plates of A are now separated to a distance na and connected one to each of the plates of B. A circular disc of radius a surrounded by a wide coplanar guard ring is placed with its plane inclined at a small angle e to another plane. The centre of the disc is at a perpendicular distance d from the plane. Four equal large conducting plates A, B, C,D are fixed parallel to one another.

A and D are connected with the earth, B has a charge E per unit area of its surface, and G a charge W. Find the potentials of B and G. A spherical conductor of radius 10 cm. Electrical connection is made, for an instant, with a distant insulated non-charged spherical conductor of radius 15 cm.

What are the final charges of the two spheres? The radii oftwo concentricconductingspheresareandcm. Calculate the mechanical stress per sq. How would these fields be altered if the spheres were connected for an instant by means of a wire? What would be the loss of electrostatic energy? Obtain expressions for the potential function in all space.

Two insulated conducting spheres, of radii 10 and 20 cm. They are so far apart that their influence on one another may be neglected. Shew that if they are joined by a wire the energy lost is equal to the work required to raise a weight of one milligram through a height of about mm.

A condenser is formed of a conducting sphere of radius r, and a conducting concentric spherical shell of internal radius B, insulated from it, the dielectric being air. Shew that, if there be placed between these two spherical conductors n conducting concentric spherical shells, the capacity becomes 1 where alt at, A circular gold leaf of radius 6 is laid on the surface of a charged conducting sphere of radius a, a being large compared to 6.

Three insulated concentric spherical conductors, whose radii in ascending order of magnitude are a, b, c, have charges elt ea, e8 respectively. Find their potentials, and shew that, if the innermost sphere be connected to earth, the potential ofthe outermost is diminished [M. A condenser is formed by a sphere B of radius b inside a con- centric sphere A of radius a.

A second condenser is formed by a sphere JD of radius d inside a concentric sphere G of radius c. Connection can be made with B, D through small holes in A, G. The sphere G is now insulated and afterwards B is joined to C by afinewire and D is earthed. The spheres A and C are so far apart that the inductive effect of either on the other may be neglected. If themiddle conductor of acondenser, formed by a long cylinder radius a inside a coaxial shell internal radius 2o and external 3a with another coaxial cylinder internal radius d outside, be charged and the other two put to earth, determine the minimum value of d in order that the parts of the middle shell may not separate, when it is cut into two parts by a plane through its axis.

A condenser is formed of three concentric cylinders of which the inner and outer are connected together. Obtain a formula for the capacity, neglecting end effects, and shew that, if the middle plate is 10 cm. On a certain day the vertical electric force in the atmosphere at the earth's surface was volts per metre and at a height of kilometres it was 25 volts per metre. Prove that the electric capacity of a conductor is less than that of any other conductor which can completely surround it.

A condenser formed of two concentric spheres of radii a and b is divided into two halves by a diametral plane, the inner and outer surfaces being rigidly connected. An electrified spherical soap bubble of radius a is surrounded by an insulated uncharged concentric spherical conducting shell whose internal and external radii are b and c; shew that when the bubble bursts the loss of electric energy is independent of c. A conducting sphere of radius r is electrified to potential V. An insulated spherical conductor, formed of two hemispherical shells in contact, whose inner and outer radii are b and b', has within it a concentric spherical conductor of radius a and without it another concentric spherical conductor of which the internal radius is c.

These two conductors are earth connected and the middle one receives a charge. An infinite circular conducting cylinder has charge q per unit length, and is surrounded by a thin coaxial cylindrical shell which is a conductor connected to earth.

If the shell is divided along two gener- ators subtending an angle 28 at the axis of the cylinders, determine the resultant force between the portions.

Calculate the capacity in fractions of a microfarad of a condenser formed by coating the inside and outside of a cylindrical j ar with tinfoil.

The radius of the jar is 12 cm. For example if a spherical conductor is slightly deformed the change in capacity is proportional to the change in volume. When an electric field is produced by charges on a number of conductors, it is clear that the distribution of the charge on each conductor is affected by inductive effects of the charges on the other conductors.

Principle of superposition. Uniqueness theorems, i There is only one vxty in which given charges can be distributed in equilibrium over a given set of conductors. Let Ax, A2, A3, If possible let there be two different ways of distributing the charges on the conductors in equili- brium; and let a, a' denote the surface densities at the same point in the two distributions.

Change the sign of the electri- fication at every point of the second distribution and then imagine the two to be superposed. This joint distribution would still be in equilibrium and it would be a distribution in which each conductor has a total charge zero.

If possible let a, a' denote surface densities at the same point in two different distributions of electricity which both cause the conductors Ax, A2, As, It follows that a distribution in which the surface density is a—a' would produce a field in which the potential of each conductor would be zero.

But the potential has no maximum or minimum in empty space, hence the potential of the charges of density a — a' is zero everywhere. Therefore there can be no electricity in the field, i. Let an electric field be produced by charges residing on a set of n conductors.

An of prescribed shapes occupying fixed relative positions in space. An are e 1; e2, We see from the definition that prs denotes the potential of the conductor As when the conductor AT has a unit charge and all the other conductors are uncharged.

Also qra is the charge induced on the conductor As when the conductor Ar is raised to unit potential and all the conductors except Ar are earth connected. Coefficients of capacity are positive, those of induction are negative and the sum of the latter belonging to any given con- ductor is generally numerically less than the coefficient of capacity of that conductor.

Therefore qn is positive. But the charges induced onA2,A3t Also the sum of the charges e2, e3, Coefficients of potential are all positive and prs cannot exceed Prr or p8S. Let Ar have a positive unit charge and let all the other conductors be uncharged, then, as in ii , pn is the highest potential in the field and is positive.

If As lies inside Ar, then, since it has no charge, its potential is the constant potential inside Ar, i. But when As lies outside Ar, then, since A3 has no total charge, as many unit tubes of force leave it as fall upon it, and those falling on it come from places of higher potential and those leaving go to places of lower potential, so that pra lies between the highest and lowest potential in the field, i.

The energy of a system of charged conductors. If it were possible to create a permanent electrostatic field in which there were no changes or movement of electricity owing to imperfect insulation or other cause, the maintenance of such a state would not require any expenditure of energy; but, in the creation of the field, work would have to be performed in the separation of the electric charges by some mechanical process.

This work is to be regarded as conserved in the field in the form of electrical energy and is available for trans- formation into heat or mechanical work when the field ceases to be static. The energy of the field of the n charged conductors of may be calculated as the work required to place the charges on the conductors.

Green's Reciprocal Theorem. Further applications of the Reciprocal Theorem, i Shew that the locus of the positions in which a unit charge will induce a given charge on a given uninsulated conductor S is an equipotential surface of 8 when 8 is freely electrified and alone in the field. Here we have two conductors, viz. S and an infinitesimal conductor at a point P.

In the first state Fig. In the second state Fig. But ex is a given charge and fa' is the potential of S when freely electrified and therefore a constant, so that fa' is constant. But fa' is the potential at P in the second state; hence the required locus of P is an equipotential surface of 8 when 8 is freely electrified and alone in the field. Hence we may suppose the thin conducting surfaces to exist in both the fields described in the problem.

In order to complete the specification of the first field Fig. We can suppose also that there is a small conductor at P which carries the charge e in the second field Fig. Applying the Reciprocal Theorem we get 1. Taking the expression for the energy from , viz. We observe that, differentiating partially, e From the foregoing quadratic expressions various theorems about the p'a and q's may be proved.

Thus taking the un- charged state of the conductors as the state of zero energy, We is essentially positive, therefore pn, p22, p33, Then by taking all the e's to be zero except ex and c2, it follows that pup22—p must be positive, and by similar arguments each of the successive discriminants Pu, Pn Pn. Pn Pn. To deduce the mechanical forces between the con- ductors from the energy. First suppose that the charges on the conductors are given and remain constant.

Wemaysup'pose that the relative positions of the conductors are determined by the values of certain co-ordinates of position x1,xi,x3, The coefficients of potential, the p'a, then depend on the x'a.

A change in any co-ordinate x will in general alter some or all of the p'a and therefore alter the energy We. Let X be the force which tends to produce a displacement 8a;. If we take as an alternative hypothesis that the potentials are kept constant while the conductors are displaced, this introduces another physical consideration, viz. In a finite displacement which only changes the co-ordinate x, in which the potentials are kept constant, the work done by the mechanical forces is i.

Thus we have mechanical work done and an equal gain in electrical energy, so that when the potentials are kept constant during a displacement, the batteries must supply an amount of energy equal to twice the mechanical work done.

We shall again use the parallel plate condenser to illustrate the theory of the preceding article. Now suppose that the plates approach one another so that their distance becomes x'. Electric screens. In order to find out what we can about the q'a take a special case: let A2 be at zero potential and A1 without charge. Then since A2 contains no charge the potential is constant i.

Hence A3 may be raised to any potential without affecting Ax and vice versa; so that the conductor A2 screens Ax from the external field. The quadrant electrometer. It consists of a metal short cylindrical box divided into four quadrants, shown in plan in the figure. Each quadrant is supported on a separate insulating stand but opposite quadrants are connected by wires.

Inside the box a flat piece of aluminium O is suspended by a silk fibre so that it can turn in a horizontal position about the axis of the cylinder. From the lower surface of G along its axis hangs a fine metal wire which connects G with the inner surface of a condenser, which is maintained at a constant high potential. The opposing pairs of quad- rants A, A' and B, B' are connected to the two bodies the difference of whose potentials is to be found, one of which might be the earth.

When A, A' and B, B' have the same potential, the needle G will be symmetrically placed with regard to them; but when the potentials of A, A' and B, B' differ, C will rotate and take upaposition of equilibrium in which the couple on it produced by the electric field inside the box is balanced by the torsion of the silk thread. We have now to determine the relation between the potentials and the angle turned through by the needle G.

Theory and Derivations are unfolded during a straightforward manner. Exercises are divided on basis of level of toughness from low to high. Hints And Complete Solutions of questions are given Separately after the chapters.

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Anurag Mishra Heat and Thermodynamics pdf. Anurag Mishra Electricity and Thermodynamics pdf. Anurag Mishra Optics pdf. Here is the final elaboration of Maxwell's theory of electromagnetism, including the systematic and rigorous derivation of his general equations of field theory.

These equations continue to occupy a central position in the modern physicist's view of the physical world. They are a magnificent summary of the fundamental advances in electricity and magnetism, and later inspired the theories of Lorentz on the electron and Einstein on relativity. Einstein himself has said that "The formulation of these equations is the most important event in physics since Newton's time" — The Evolution of Physics.