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Magnetostatics There are two major laws governing magnetostatics: 1. Biot-Savart law 2. Ampere’s circuit law Just as Gauss’s law is special case of Coulombs law, Ampere’s law is a special case of Biot-Savart’s law. We know, Coulomb's law relates electric fields to the point charges which are their sources. In a similar manner, BiotSavart Law relates magnetic fields to the currents which are their sources. 1 Derivation of Gauss’ Law using Coulombs law Consider a sphere drawn around a positive point charge. Evaluate the net flux through the closed surface. Net Flux = E dA E cos dA EdA kq For a Point charge E 2 r kq EdA 2 dA r kq kq 2 dA 2 (4 r 2 ) r r 4 kq 4 k 1 0 net where 0 8.85x10 qenc 0 E n cos 0 1 nˆ dA 12 Gauss’ Law dA nˆdA C2 Nm 2 2 Biot-Savart Law An electric current produces a magnetic field. Consider a small segment of wire with length dl carrying a current I. This small segment will produce a small magnetic field dB at a point P whose magnitude and direction are given by the Biot-Savart law. o I dl * r̂ dB 4 r 2 r̂ is a unit vector Specifying the direction of r ( distance from the current to the field) and µ0 is the permeability. The Biot-Savart Law is used to calculate the magnetic field at a given position. 3 2. Gauss’ law for Magnetic Fields Gauss’ law for magnetic fields is a formal way of saying that magnetic monopoles do not exist. The law states that the net magnetic flux B through any closed surface must be zero. B dS 0 B ** Maxwell’s second equation For comparison, Gauss’ law for electric fields E dS E Qenc 0 In both equations, the integral is taken over a closed Gaussian surface. E o 4 Ampere’s Circuital Law in Integral Form • It states that “the circulation of the magnetic flux density in free space over a closed path is proportional to the total current enclosed in the surface.” B dl I 0 encl C Iencl is current through S: I encl J dS S B dl J dS 0 C S Where J is defined as current density8 Ampere’s Circuital Law in Differential form Applying Stroke's theorem to left-hand side of above equation B dl B dS J dS 0 L S S Comparing the surface integrals in above expressions B 0 J Note: ** Ampere’s circuital Law is Maxwell’s IV equation. H J 0 This shows that magnetostatic field is nonconservative in nature. 9 3. Faraday’s Law ** Changing magnetic field gives rise to electric current. Induced emf if the loop is open-circuited B t Loop E EMF(Ve) B (increasing) Induced emf in the loop is > E dl P ^ I < Field by I Integral form Differential form B P E d l t B E t Note: ** Faraday’s Law is Maxwell’s III equation. B (increasing) ^ Four fundamental Laws: 1. 2. E dS S Qenclosed o B dS 0 S 3. 4. B E dl t P B dl 0 I encl Faradays Amp Law P Asymmetry in the above laws 1. In equation 1 and 2, the R.H.S. of equation 2 contains no value, because of the non existence of the magnetic monopole. 2. In equation 3 changing magnetic flux gives rise to electric field. There is no such corresponding term relating to changing electric flux producing magnetic field in equation 4. This led Maxwell to introduce a new term corresponding to changing electric flux in equation 4. 11 The corresponding term is 0 0 E dl E t B dl 0 I encl 0 0 P P E 0 ( I encl I d ) t I d displacement curent 0 B t B dl 0 I encl P E t Maxwell’s Equations in Integral Form 1. Qenclosed E d S o 3. P S 2. B dS 0 E dl 4. B t E B d l I 0 P 0 encl t S 12 CONTINUITY EQUATION From the principle of charge conservation, the rate of decrease of charge within a given volume must be equal to the net current flowing out through the surface of the volume. Thus current Iout coming out of the closed surface is dQin I out J dS dt where Qin is the total charge enclosed by the closed surface. Using the divergence theorem J dS Jdv S and v dQin d v v dv dv dt dt v v t v dv Jdv v v t v J t 13 This equation is called continuity equation. It is derived from principle of conservation of charge and shows that there can not be accumulation of charge at any point. For steady currents, v 0 t J 0 Hence the total charge leaving a volume is the same as the total charge entering it. 14 Maxwell’s modification of Ampere’s Law In Differential Form B dl 0 I encl From Ampere’s Law where P I encl J dS S Applying Stroke's theorem to left-hand side of above equation B dS 0 J dS S Differential form of Ampere’s Law S B 0J 15 Taking divergence of above equation B 0 0 J Since R.H.S. of above equation is not zero**, so let us apply Continuity equation and Gauss’s law (Electrostatics) we get v E J 0 E 0 t t t E J 0 0 t Hence E or . J 0 0 t B 0 (J 0 E ) t E E The term 0 is called as displacement current density . J d 0 t t Hence modified Ampere’s Law (differential form) B 0 J J d **Note: Fundamental theorem of vector analysis; B 0 16 Maxwell’s Equations in Integral Form 1. 2. E dS S Qenclosed o B dS 0 S 3. 4. “Gauss Law in Electrostatics”; Electric flux coming out from the surface of the body is equivalent to the charge enclosed by the body. “Gauss Law in Magnetostatics”; Magnetic flux coming out from the closed surface of the body is zero as no magnetic monopole exist. “Faradey’s Law”; Changing Magnetic flux produce electric current (or field) in aclosed loop. where Magnetic flux B B.ds B E dl t L E B dl 0 i 0 t L S “Modified Ampere’s Law” or “MaxwellAmpere’s Law”; Changing electric flux can produce magnetic field in a discontinuous circuit to hold Ampere’s circuital Law. where Electric flux E E.ds S 17 Maxwell’s Equations in Integral Form 1. Qenclosed E d S o 3. P S 2. E dl B dS 0 4. B t E B dl 0 i 0 t P S Maxwell’s Equations in differential form 1. E 2. 0 B 0 3. 4. E - B t B 0 (J 0 E ) t 20 Case 1: Maxwell equations in free space* (no free charges and no currents) 0, J 0, q 0, i 0 E 0 B 0 B E t E B 0 0 t * Helpful to understand Electromagnetic waves in free space. 21