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GANDHINAGAR INSTITUTE OF TECHNOLOGY Electromagnetic Active Learning Assignment 130120111010 Mihir Hathi 130120111012 Komal Kumari Guided By: Prof. Mayank Kapadiya 1 Branch : Electronic and Communication Engineering Div: E CONTENT 1. 2. 3. 4. 5. 6. 7. Biot-Savart Law Introduction Set-Up Observations Equation Total Magnetic Field Final Notes Special Cases 2 Magnetic Flux Gauss’ Law in Magnetism Ampere’s Law – General Form STOKES’ THEOREM 3 BIOT-SAVART LAW 4 INTRODUCTION Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet They arrived at a mathematical expression that gives the magnetic field at some point in space due to a current 5 SET-UP The magnetic field is dB at some point P The length element is ds Please replace with fig. 30.1 The wire is carrying a steady current of I 6 OBSERVATIONS The vector dBis perpendicular to both dsand to the unit vector r̂ directed from dB toward P The magnitude of ds is inversely proportional to r2, where r is the distance from dsto P The magnitude of dBis proportional to the current and to the magnitude ds of the length element ds The magnitude of dBis proportional to sin q, where q is the angle between the vectors r̂ and ds 7 EQUATION The observations are summarized in the mathematical equation called the Biot-Savart law: μo I ds ˆr dB 4π r2 The magnetic field described by the law is the field due to the current-carrying conductor Don’t confuse this field with a field external to the conductor 8 PERMEABILITY OF FREE SPACE The constant mo is called the permeability of free space mo = 4p x 10-7 T. m / A 9 TOTAL MAGNETIC FIELD dBis the field created by the current in the length segment ds To find the total field, sum up the contributions from all the current elements I ds μo I ds ˆr B 4π r 2 The integral is over the entire current distribution 10 FINAL NOTES The law is also valid for a current consisting of charges flowing through space dsrepresents the length of a small segment of space in which the charges flow For example, this could apply to the electron beam in a TV set 11 BFOR A LONG, STRAIGHT CONDUCTOR The thin, straight wire is carrying a constant current ds ˆr dx sin θ k̂ Integrating over all the current elements gives μo I θ2 B cos θ dθ θ 4πa 1 μo I sin θ1 sin θ2 4πa 12 B FOR A LONG, STRAIGHT CONDUCTOR, SPECIAL CASE If the conductor is an infinitely long, straight wire, q1 = p/2 and q2 = -p/2 The field becomes μo I B 2πa 13 BFOR A CURVED WIRE SEGMENT Find the field at point O due to the wire segment I and R are constants μo I B θ 4πR q will be in radians 14 SPECIAL CASES 15 BFOR A CIRCULAR LOOP OF WIRE Consider the previous result, with a full circle q = 2p μo I μo I μo I B θ 2π 4πa 4πa 2a This is the field at the center of the loop 16 BFOR A CIRCULAR CURRENT LOOP The loop has a radius of R and carries a steady current of I Find the field μo I at a 2point P Bx 2 a x 2 2 3 2 17 MAGNETIC FLUX The magnetic flux associated with a magnetic field is defined in a way similar to electric flux Consider an area element dA on an arbitrarily shaped surface 18 The magnetic field in this element is B dA is a vector that is perpendicular to the surface dA has a magnitude equal to the area dA The magnetic flux FB is B B dA The unit of magnetic flux is T.m2 = Wb Wb is a weber In which the magnetic flux density (or magnetic induction) in free space is: and where the free space permeability is 19 GAUSS’ LAW IN MAGNETISM Magnetic fields do not begin or end at any point The number of lines entering a surface equals the number of lines leaving the surface Gauss’ law in magnetism says the magnetic flux through any closed surface is always zero: B d A 0 20 AMPERE’S CIRCUITAL LAW In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère in 1826,relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it again using hydrodynamics in his 1861 paper On Physical Lines of Force and it is now one of the Maxwell equations, which form the basis of classical electromagnetism. 21 Where J is the total current density (in ampere per square metre, Am−2). 22 where Jf is the free current density only. Furthermore • is the closed line integral around the closed curve C, • denotes a 2d surface integral over S enclosed by C •dℓ is an infinitesimal element of the curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C) •dS is the vector area of an infinitesimal element of surface S STOKES’ THEOREM The theorem is named after the Irish mathematical physicist Sir George Stokes (1819–1903). What we call Stokes’ Theorem was actually discovered by the Scottish physicist Sir William Thomson (1824–1907, known as Lord Kelvin). Stokes learned of it in a letter from Thomson in 1850. 24 LET S be an oriented piecewise-smooth surface bounded by a simple, closed, piecewise-smooth boundary curve C with positive orientation F be a vector field whose components have continuous partial derivatives on an open region in R3 that contains S. Then, ∫ F ⋅ dr = C S ∫∫ curl F ⋅dS Thus, Stokes’ Theorem says: The line integral around the boundary curve of S of the tangential component of F is equal to the surface integral of the normal component of the curl of F. 25 26