Engineering Physics L-1 PDF

Engineering Physics L-1 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Tha...

Engineering Physics L-1 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Thakur SOLID STATE PHYSICS Lovely Professional University Phagwara, Punjab-144411 FIBRE OPTICS EMT-Topics 1. Scalar and Vector fields. 2. Concepts of gradient, divergence and curl. 3. Gauss Theorem and Stokes theorem. 4. Poisson and Laplace’s Equation. 5. Continuity equation. 6. Maxwell’s Electromagnetic equation. 7. Physical significance of Maxwell’s equation. 8. Ampere’s Circuital law. 9. Maxwell displacement current and correction in Amperes law. Scalars and Vectors Scalars and Vectors In Which Direction ? What is the temperature? What is the Distance covered? What is the speed ? How much energy does he spend? What is the velocity ? What is his height? Is there any force on him ? What is his weight? What is his mass? How is it that he is on the ground ?  Scalars and Vectors Scalars Vectors Vectors Mass (Kg) Force Vectors have both magnitude and Temperature (K) Velocity direction. Current (A) Displacement Ways of writing vector notation F ma Speed (m/s) Fields (Electric and   Magnetic) F ma Time (s) Acceleration F ma Amount of Substance Current density (J=I/A) (mol) Luminous Intensity (cd) Length Energy… Momentum… Vectors: The vectors associated with the linear directional effect are called the polar vectors. Examples ar momentum, linear velocity, etc. Vectors: The vectors associated with the rotation about an axis are called the axial vectors. Examples are ar velocity, angular momentum etc. Vector Math   Triangular law of vector addition: Vector Inverse A  A If two vectors acting simultaneously on Just switch direction a body are represented both in   magnitude and direction by two sides A B   of a triangle taken in an order then the Vector Addition A B A resultant (both magnitude and Use head-tail method, or    B direction) of these vectors is given by parallelogram method  A B third side of that triangle B taken in A  opposite order. Vector Subtraction A   A B Use inverse, then add    A B Parallelogram law of vector  B addition: Vector Multiplication If two vectors are considered to be the Two kinds! adjacent Scalar, or dot product sides of a Parallelogram, then the Vector, or cross product resultant of two vectors is given by the vector which ⃗is+⃗𝐵 diagonal passing a Vector Addition by Components A the 𝐴   through ⃗𝑅 =point  of contact of two A  B ( Ax  Bx )iˆ  ( Ay  B y ) ˆj  ( Az  Bz )kˆ vectors. 𝜃 B Vector Math The magnitude of the resultant is given by  B P  ⃗  A 𝑅 A 𝐴 𝑠𝑖𝑛 𝜃 𝜃 𝜃 Q 𝑅=√ 𝐴 +𝐵 +2 𝐴𝐵𝑐𝑜𝑠 𝜃 𝛽 2 2  O B 𝐴 𝑐𝑜𝑠 𝜃 The angle between the resultant and the horizontal vector Can be obtained by extending the horizontal vector as shown in The figure. Thus, for the triangle OQP, 𝛽=tan − 1 ( 𝐴 𝑠𝑖𝑛 𝜃 𝐵+ 𝐴 𝑐𝑜𝑠 𝜃 ) Representation of a vector, Unit Vector and Rectangular Resolution of vectors y  A vector may be represented as B   P ry r Here, is the vector, is the magnitude of the vector q  O rx A x and is the unit vector.  Thus, r gives the magnitude and gives the From the right angled triangle OAP we may write the magnitude of the given direction of the vector. vector, We may then write.  Unit vectors have unit magnitude but a definite direction.  Let OX and OY be the two rectangular axes. Let and are the two unit vectors along x and y respectively.  If and the coordinates of P be (x,y), then, From the same triangle OAP and.  Again by the law of vector addition (For convenience bold face will be considered as vector notation) Or, This is known as rectangular resolution of vectors. Test Q. If a vector in 3D is given as What will be the unit vector? Ans. Product of two Vectors: Dot Product Scalar Product or Dot product   Q. How do we show that A B  Ax Bx  Ay B y  Az Bz The dot product says  something about how Let, A  Ax iˆ  Ay ˆj  Az kˆ parallel two vectors are.  B Bx iˆ  B y ˆj  Bz kˆ The dot product (scalar product) of two vectors can be thought of as the The dot product of perpendicular unit vector is zero. projection of one onto the  The dot product of parallel unit vector is unity.   B   direction A B of ABthe cos  other. A B  Ax iˆ Bx iˆ  Ay ˆj B y ˆj  Az kˆ Bz kˆ  A iˆ  A cos   Ax ( 𝐴𝑐𝑜𝑠 𝜃 ) 𝐵  Ax Bx  Ay B y  Az Bz  q A In the component form 𝐴(𝐵𝑐𝑜𝑠 𝜃)  A B  Ax Bx  Ay B y  Az Bz **Scalar product of two vectors is a scalar. Product of two Vectors: Vector Product Vector Product or Cross product  Recall angular momentum z    L r p The cross product of two  Torque vectors says something    about how perpendicular  r F they are. You will find it in **Vector product of two vectors is a vector the context of rotation, or y twist.   A B  AB sin  Cross of unit vectors x ⃗ 𝐴× ⃗ 𝐵 Direction perpendicular to both A and  B (right-hand   rule) A B  B A ⃗ 𝐵 iˆ ˆj kˆ   ˆ ˆ ˆ A B  Ax Ay Az ( Ay Bz  Az B y )i  ( Az Bx  Ax Bz ) j  ( Ax B y  Ay Bx )k 𝜃 Bx By Bz ⃗ 𝐴 Differentiation of Vectors 𝑑 𝑑𝑩 𝑑 𝑨 ( 𝑨 × 𝑩 )= 𝑨× + ×𝑩 𝑑𝑠 𝑑𝑠 𝑑𝑠 𝑑𝑪 𝑑𝑠 =0 ^ 𝐹 ^𝑗+ 𝐹 𝑘 𝑭 = 𝐹 𝑥 𝑖+ ^ 𝑦 𝑧 𝑑 𝑑𝑨 ( 𝑎 𝑨 )=𝑎 𝑑𝑠 𝑑𝑠 𝑑 𝑑𝑽 𝑑𝑈 ( 𝑈 𝑽 )=𝑈 +𝑽 𝑑𝑠 𝑑𝑠 𝑑𝑠 Differential operator: (nabla) is designated as the differential vector operator and is given by, GRADIENT (), DIVERGENCE() AND CURL()  GRADIENT (), DIVERGENCE() AND CURL() A continuous function of position of a point in a region is called a point function. The region of space in which It specifies a physical quantity is known as a field. These fields are classified into two groups: Scalar Field: It is defined as that region of space where each point is associated with a scalar point function. That is a continuous function which gives the value of a physical quantity such as, temperature, potential etc. In a scalar field, all the Temperature field or a scalar field points having the same scalar physical quantity are connected by the means of surfaces called equal or the level surfaces. Vector Field: A vector field is specified by a continuous vector point function having magnitude and direction and both of which may change from point to point in a given region of field. The field is represented by vector lines or lines of surfaces. The tangent at a vector line gives the direction of theMagnetic field of bar magnet Earth’s magnetic field vector at that point. Gradient () 𝜕 𝜙 ^ 𝜕 𝜙 ^ 𝜕𝜙 ^ dr 𝛁 𝝓= 𝑖+ 𝑗+ 𝑘 𝜕𝑥 𝜕𝑦 𝜕𝑧 r P(x,y,z) Physical Meaning: The gradient points in the direction of maximum increase of. || gives the slope (rate of increase) along this maximal direction. Gradient can be used to find the Gradient () d 𝜙=𝜵 𝝓. 𝒅𝒓 =|𝛻 𝜙||𝑑𝑟|cos ⁡( 𝜃)  From the above equation it is evident that if we fix ‘dr’ and vary the then maximum change in the function can happen only if i.e. if we move in the same direction as the direction of the gradient.  Imagine you are standing on a hillside. Look all around you, and find the direction of steepest ascent. That is the direction of the gradient. The function however this time is the height. Gradient () What would it mean for the gradient to vanish? If = 0 at (x, y, z) then = 0 for small displacements about the point (x, y, z). This is, then, a stationary point of the function (x, y, z). It could be a maximum (a summit), a minimum (a valley), a saddle point (a pass). This is analogous to the situation for functions of one variable, where a vanishing derivative signals a maximum, a minimum, or an inflection. In particular, if you want to locate the extrema of a function of three variables, set its gradient equal to zero. Gradient (): Checkpoint Q1. Find the directional derivative of the function at the point (1,2,-1). Q2. Find the directional derivative of the function at the point (1,2,-1) in the direction. ⃗ 𝛻𝜙 𝜃 i rect i on o f d in the Gradient (): Checkpoint Divergence () The name divergence is well chosen, for is a measure of how much the vector v spreads out (diverges) from the point in question. For example, the vector function in Fig. a has a large (positive) divergence (if the arrows pointed in, Observe that the divergence of a vector function v is it would be a negative divergence). itself is a scalar. The function in Figure (b) has zero divergence. The function in Fig. c again has a positive divergence. (Please understand that v here is a function—there's a different vector associated with every point in space. In the diagrams, of course, we can only draw the arrows at a few Curl () Curl () The name curl is also well chosen, for is a measure of how much the vector v swirls around the point in question. The functions in Figures have a substantial curl, pointing in the z direction, as the natural right-hand rule would suggest. Imagine (again) you are standing at the edge of a pond. Float a small paddlewheel (a cork with toothpicks pointing out radially would do); if it starts to rotate, then you placed it at a point of nonzero curl. A whirlpool would be a region of large  Important to Note the divergence of a vector field is zero, we call that to be SOLENOIDAL or INCOMRESSIBLE. the curl of a vector field is zero then it is called IRROTATIONAL and CONSERVATIVE. Prove?  Divergence of Curl is Zero  Curl of a Gradient is zero  Engineering Physics L-2 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Thakur SOLID STATE PHYSICS Lovely Professional University Phagwara, Punjab-144411 FIBRE OPTICS EMT-Topics 1. Scalar and Vector fields. 2. Concepts of gradient, divergence and curl. 3. Gauss Theorem and Stokes theorem. 4. Poisson and Laplace’s Equation. 5. Continuity equation. 6. Maxwell’s Electromagnetic equation. 7. Physical significance of Maxwell’s equation. 8. Ampere’s Circuital law. 9. Maxwell displacement current and correction in Amperes law. Recap Divergence: Curl: Gradient: Imagine a fluid, with the vector field Let's go back to our fluid, Gradient tells you how much representing the velocity of the fluid with the vector field something changes as you mov at each point in space. Divergence representing fluid from one point to another. measures the net flow of fluid out of velocity. The curl (i.e., diverging from) a given point. If measures the degree to fluid is instead flowing into that point, which the fluid is rotating the divergence will be negative. about a given point, with whirlpools and tornadoes A point or region with positive being extreme examples. divergence is often referred to as a "source" (of fluid, or whatever the field is describing), while a point or region with negative divergence is a "sink" https:// Recap At each point, the gradient is a vector which points in the direction at which the scalar field rises the most rapidly, and the length of the vector corresponds to the steepness of the rise. If you're at the very top of the hill, there's no going up. The gradient here is 0. Similarly, if you're at the very bottom of the valley, every direction is up; the gradient here is also 0. But in all other places (assuming the terrain is smooth, not ragged with sheer cliffs and fissures), there's a single direction which points in the steepest up route, and that's where the gradient points at. https://www.quora.com/How-do-we-visualise-gradient-vector-of-a-scalar-field  Recap: Important to Note the divergence of a vector field is zero, we call that to be SOLENOIDAL or INCOMPRESSIBLE. the curl of a vector field is zero then it is called IRROTATIONAL and CONSERVATIVE. Prove? ( ^ 𝛁. (𝛁 × 𝒗 )= 𝑖 𝜕 ^ 𝜕 ^ 𝜕 𝜕𝑥 +𝑗 𝜕𝑦 +𝑘 𝜕𝑧 ). (𝛁 × 𝒗 )  Divergence of Curl is Zero  Curl of a Gradient is zero | | 𝑖^ ^𝑗 ^ 𝑘 𝜕 𝜕 𝜕 𝛁 × (𝛁 𝑺)= 𝜕 𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑆 𝜕𝑆 𝜕𝑆 𝜕𝑥 𝜕𝑦 𝜕𝑥 Line Integral, Surface Integral and Volume Integral Line Integral where v is a vector function, dl is the infinitesimal displacement vector, and the integral is to be carried out along a prescribed path V from point a to point b. If the path in question forms a closed loop (that is, if b = a), then we write, Line Integral  Ordinarily, the value of a line integral depends critically on the path taken from a to b, but there is an important special class of vector functions for which the line integral is independent of path and is determined entirely by the end points. Such vector fields are known as conservative fields.  In physics if a force has this property then we call that force conservative.  Gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force.  Other examples of conservative forces are: force in elastic spring, electrostatic force between two electric charges, magnetic force between two magnetic poles. Surface Integral Where, v is again some vector function, and the integral is over a specified surface S. Here da is an infinitesimal patch of area, with direction perpendicular to the surface as shown in figure. If the surface is closed (forming a "balloon"), then again put a circle on the integral sign. J 𝐼=∫ 𝑱.𝒅𝒂 da Volume Integral Fundamental Theorems of Calculus Three fundamental theorems associated with each of the three quantities gradient, divergence and curl. The fundamental theorem of divergence is also known as Gauss theorem or Green’s theorem. The fundamental theorem of curl is also known as Stokes Theorem. Fundamental Theorem of Gradient  Let us consider a function T(x, y, z).  The function ‘T’ may represent any scalar quantity like temperature, potential etc.  Starting at any point a, we move a small distance. According to the definition of gradient, the function T will change by an amount Fundamental Theorem of Gradient Fundamental Theorem of Gradient Fundamental Theorem of Divergence or Gauss Theorem It states that, the integral of the divergence over a region of space (a volume, V), is equal to the value of the function at the boundary (the surface S that bounds the volume). * Boundary of a volume is a surface If v represents the flow of an incompressible fluid, then the flux of v (the right side of the above Equation) is the total amount of fluid passing out through the surface, per unit time. Now, the divergence measures the "spreading out" of the vectors from a point—a place of high divergence is like a "faucet," pouring out liquid. If we have a bunch of faucets in a region filled with incompressible fluid, an equal amount of liquid will be forced out through the boundaries of the region. Fundamental Theorem of Curl or Stokes Theorem It states that, the integral of the curl over a patch of surface, S is equal to the value of the function at the boundary (here, the perimeter of the patch, V). As in the case of the divergence theorem, the boundary term is itself an integral—specifically, a closed line integral. Fundamental Theorem of Curl or Stokes Theorem Fundamental Theorem of Curl or Stokes Theorem  Recall that the curl measures the "twist" of the vectors v.  A region of high curl is a whirlpool—if you put a tiny paddle wheel there, it will rotate.  Now, the integral of the curl over some surface (or, more precisely, the flux of the curl through that surface) represents the "total amount of swirl," and we can determine that just as well by going around the edge and finding how much the flow is following the boundary.  Engineering Physics L-3 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Thakur SOLID STATE PHYSICS Lovely Professional University Phagwara, Punjab-144411 FIBRE OPTICS EMT-Topics 1. Scalar and Vector fields. 2. Concepts of gradient, divergence and curl. 3. Gauss Theorem and Stokes theorem. 4. Poisson and Laplace’s Equation. 5. Continuity equation. 6. Maxwell’s Electromagnetic equation. 7. Physical significance of Maxwell’s equation. 8. Ampere’s Circuital law. 9. Maxwell displacement current and correction in Amperes law.  Coulomb’s Law, Electric Field and Electric Potential oulomb’s Law F +q +q F he force acting between two charged particles is given by 1 r 2 ke charges repel each other and unlike charges attract each other. Electric Field Electric field is the region over which the effect of a charge can be felt. r If another charge appears in this field, it will be either + repelled or attracted based on the nature of the charge q (i.e. +ve or -ve). Physical expression of the electric field is given by, If ‘q’ is +ve, then field extends outward from the charge and if it is negative the field will extend inward into the charge.  Coulomb’s Law, Electric Field and Electric Potential + -q q +q Electric Potential Positive Field 𝑏=∞ Negative Field An electric potential is the amount of work needed to move a unit positive charge from a reference point to a specific point inside the field. Typically, the reference point is Earth or a point at Infinity, although any point beyond a the influence of the electric field can be used for this purpose. Electric potentials are the line integrals of the electric field + over some region. q and e gradient of potential is in the opposite direction to the electric field. Electric field and potential for a charge distribution Electric potential due to a point charge at a distance r is given by For a volume charge distribution with density the electric field is expressed as + + + + + + + Where + r’ + + + ++ If the charge density is not uniform within the sphere then will be a function + + + + + of the radius of the sphere (r’). Thus. + + + r Unfortunately, integrals of this type can be difficult to calculate for any but the simplest charge configurations. Occasionally, we can get around this by exploiting the symmetry and using Gauss's law, but ordinarily the best strategy is first to calculate the potential, V, which is given by,  Electric flux and Gauss’s Law Electric flux: It is defined as the number of electric force field lines crossing a given area in a direction perpendicular to the area. The electric flux through an elementary area is given as The total flux through any finite area is obtained as Gauss’s Law: It states that the total electric flux through a closed surface is equal to the times the charge enclosed by the surface. and Poisson and Laplace equations We have determined the electric field 𝐸 in a region using Coulomb’s law or Gauss law when the charge distribution is using the relation 𝐸 = -𝛻𝑉 when the potential V is specified specified in the region or throughout the region. However, in practical cases, neither the charge distribution nor the potential distribution is specified only at some boundaries. These type of problems are known as electrostatic boundary value problems. For these type of problems, the field and the potential V are determined by using Poisson’s equation or Laplace’s equation. Laplace’s equation is the special case of Poisson’s equation. Poisson and Laplace equations For the Linear material Poisson’s and Laplace’s equation can be easily derived from Gauss’s equation But therefore This equation is known as Poisson’s equation which state that the potential distribution in a region depend on the local charge distribution. In many boundary value problems, the charge distribution is involved on the surface of the conductor for which the free volume charge density is zero, i.e., =0. In that case, Poisson’s equation reduces to, This equation is known as Laplace’s equation. Applications of Poisson and Laplace equations Application of Laplace’s and Poisson’s Equation Using Laplace or Poisson’s equation we can obtain: 1. Potential at any point in between two surface when potential at two surface are given. 2. We can also obtain capacitance between these two surface. Poisson Equation In such cases it is reasonable to recast the problem into the following form This equation is known as the Poisson’s equation Laplace Equation If in a region the charge Q. Obtain an expression for the potential along x-axis having the conditions that V=4 density is zero, then the at x=1 and V=0 at x=5. Poisson’s equation is modified as This is known as the Laplace equation. In 3 Dimension this can be written as Continuity Equation  A continuity equation in physics is an equation that describes the transport of some quantity.  Flow of charge per unit time is designated as current (I).  Again flow or charge per unit time per unit area is known as the flux (). Or,  This above quantity is defined as the current density (J). It is defined as the current per unit area through a surface. Equation of Continuity says, in a volume , if there is a divergence of the current density (J) (or the flux) within the volume, then it implies that the charges are flowing out through the volume. Thus, the negative rate of change of charges in a volume is equal to the divergence of the current density. Mathematically, Or, This is known as the equation of continuity. Continuity Equation: Derivation If charges are flowing out of a volume V, then the electric current Using Gauss divergence theorem on the LHS is given by we get The negative sign appears because charges decreases Thus, in volume due to outflow. Or, If dq charge is enclosed in a volume element and is leaving 𝜕𝜌 𝛻. 𝐽 =− a surface of area dS then we have, 𝜕𝑡 And Continuity equation says that total current flowing out of some volume must be equal to the rate of decrease of the charge within that Thus, the above equation is written as volume (if the charge is neither being created nor being destroyed). The transport of charge constitutes the current. In case of stationary current and we have That is there is no net outward flux of current density J.  Engineering Physics L-4 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Thakur SOLID STATE PHYSICS Lovely Professional University Phagwara, Punjab-144411 FIBRE OPTICS EMT-Topics 1. Scalar and Vector fields. 2. Concepts of gradient, divergence and curl. 3. Gauss Theorem and Stokes theorem. 4. Poisson and Laplace’s Equation. 5. Continuity equation. 6. Maxwell’s Electromagnetic equation. 7. Physical significance of Maxwell’s equation. 8. Ampere’s Circuital law. 9. Maxwell displacement current and correction in Amperes law. Maxwell’s Electromagnetic Equation Ampere’s Circuital Law: Proof Ampere’s Law: Gauss Law:  The law is analogous to gauss law in The net electric flux through any electrostatics. closed surface drawn in an electric  It sates that the line integral of magnetic field is equal to times the total field around any closed loop is equal to charge enclosed within the surface. (permeability of free space) times the net current flowing through the area enclosed by the loop. The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb's law relates electric fields to the point charges which are their sources. Ampere’s Circuital Law Ampere’s Law We know that if there is a steady current flowing through a wire, then a magnetic field is found to encircle the wire in concentric circle. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. But, Ampere’s Law Therefore, Using Stokes theorem Thus, Current coming out of the page 𝛁 × 𝑩=𝜇 0 𝑱 Ampere’s Circuital Law: Proof Proof: According to the Biot-Savart law, the magnitude of the magnetic field at a point P and at a distance r from the I conductor carrying current I is ∮ 𝑩.𝒅𝒍=∮ 𝐵𝑑𝑙𝑐𝑜𝑠𝜃 B dl C r But, between B and dl is zero, hence, ∮ 𝑩.𝒅𝒍=∮ 𝐵𝑑𝑙=𝐵∮ 𝑑𝑙 Again ∮ 𝑑𝑙=2𝜋𝑟 Therefore, Maxwell’s Modification to ampere’s law Before Maxwell started his work, the laws of Electromagnetism looked like below. However, they were not used to be written in the following compact form. They used to exist independently. Maxwell noticed an inconsistency in the fourth equation. Let us see what is that  Maxwell’s Modification to ampere’s law be noted that divergence of a curl is zero. If we take the divergence of the fourth equation then w (1) The left side is zero by the argument divergence of a curl is zero. But the right hand side is not zero. This is because from the continuity equation, (2) The problem is on the right side of the equation (1). If we apply Gauss law, we get, Thus, Maxwell’s Modification to ampere’s law Therefore, 𝝏𝑬 𝛁 × 𝑩=𝝁𝟎 ( 𝑱 + 𝝐𝟎 ) 𝝏𝒕 ( 𝛁. 𝑱 +𝝐 𝟎 𝝏𝑬 𝝏𝒕 ) =𝟎 Total Current density () Now, if we rewrite the Ampere’s law as Then, upon taking divergence on both the sides we get, This time both the sides are zero. Maxwell’s Modification to ampere’s law 𝝏𝑬 𝛁 × 𝑩=𝝁𝟎 ( 𝑱 + 𝝐𝟎 ) 𝝏𝒕 Displacement current has nothing to do with real current, except that it adds to J in Ampere's law.  Engineering Physics L-5 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Thakur SOLID STATE PHYSICS Lovely Professional University Phagwara, Punjab-144411 FIBRE OPTICS EMT-Topics 1. Scalar and Vector fields. 2. Concepts of gradient, divergence and curl. 3. Gauss Theorem and Stokes theorem. 4. Poisson and Laplace’s Equation. 5. Continuity equation. 6. Maxwell’s Electromagnetic equation. 7. Physical significance of Maxwell’s equation. 8. Ampere’s Circuital law. 9. Maxwell displacement current and correction in Amperes law. Displacement Current: Characteristics of displacement current: 1. It has the same unit as electric current density. 2. Displacement current is a current only in the sense that it produces magnetic field. It has none other properties of current. For example displacement current can have a finite value in vacuum where there are no charges of any type. 3. It serves to make the total current continuous across discontinuities in conduction current. 4. It makes the Ampere’s circuital law a consistent one. Maxwell came up with this rather curious name because many of his ideas regarding electric and magnetic fields were completely wrong. For instance, Maxwell believed in the æther, and he thought that electric and magnetic fields were some sort of stresses in this medium. He also thought that the displacement current was associated with displacements of the æther (hence, the name). The reason that these misconceptions did not invalidate his equations is quite simple. Maxwell based his equations on the results of experiments, and he added in his extra term so as to make these equations mathematically self-consistent. Both of these steps are valid irrespective of the existence or non-existence of the æther.  Displacement Current: Capacitor example ow can you say that for steady current, the divergence inconsistency is removed? One more way the Ampere’s law fails for non- steady current, other than the divergence inconsistency is while charging a capacitor. How to determine the current ? Well, it is the total current passing through the loop or the current piercing through the surface that has the loop for its boundary. In this case the simplest surface lies in the plane of the loop and the wire punctures the surface. Thus,. But, if we draw a balloon shaped surface, then no current is passes through the surface and hence. Thus based on the surface selected for the Amperian loop, the results for enclosed currents are different and accordingly the circulation of magnetic field will also be affected. Maxwell’s Equation: Differential Form Some people call this as Gauss law in magneto statics. Maxwell’s Equation: Integral Form Physical Significance of Maxwell’s Equation  The equations represent the fundamentals of electricity and magnetism in concise form.  The total flux of electric field through a closed surface is equal to () times the total charge enclosed within the volume.  The net magnetic flux emerging through any closed surface is zero.  The line integral of electric field around a closed path is equal to the rate of change of magnetic flux through any surface bounded by the path.  The line integral of magnetic field around a closed path is equal to the net flux of density of electric current and displacement current.  Engineering Physics L-6 ELECTROMAGNETIC THEORY (Electromagnetic Theory) QUANTUM MECHANICS LASER WAVES Dr. Vishal Thakur SOLID STATE PHYSICS Lovely Professional University Phagwara, Punjab-144411 FIBRE OPTICS Dielectrics  Dielectrics are insulators which do not conduct electricity. However, based on the function they perform, their nomenclature is different.  When the main function of non-conducting materials is to provide electrical insulation, they are called insulators.  On the other hand, non conducting materials when placed in an electric field, modifies the electric field and themselves undergo appreciable changes. As a consequence, they act as electrical charge stores.  Thus, when charge storage is the main function of the of the non conducting material, they are called dielectrics.  Dielectric Constant  The dielectric constant characterizes a dielectric material.  It is also called the relative permittivity. + + ++ + + + + ++ + + + If V is the voltage applied across ++ + + - parallel plate capacitor, then charges - on one plate will be +Q and on the - - - - - other plate will be –Q. The capacitance - - - - -- C is related by the quantity of charge stored on either plate. Thus, If the plates have area A and a vacuum in between then the capacitance is written as, Here, is called the permittivity of free space or vacuum. This is a universal constant with value. Dielectric Constant Let us insert a piece of dielectric material inside the plates. The Dielectric material capacitance can be written as Where, is the permittivity of this dielectric medium and. The dielectric constant of the inserted material is thus defined as It is a dimensionless quantity and it is independent of the size and shape of the dielectric. This is also called the relative permittivity of the material. For dielectrics, Electric Field in Matter P is called the polarization.