1. G ANDHINAGAR I NSTITUTE OF T ECHNOLOGY Electromagnetic Active Learning Assignment 130120111010 Mihir Hathi 130120111012 Komal Kumari Guided By: Prof. Mayank Kapadiya Branch : Electronic and Communication Engineering Div: E 1

2. C ONTENT Biot-Savart Law 1. Introduction 2. Set-Up 3. Observations 4. Equation 5. Total Magnetic Field 6. Final Notes 7. Special Cases 2

3. Magnetic Flux Gauss’ Law in Magnetism Ampere’s Law – General Form STOKES’ THEOREM 3

4. B IOT -S AVART L AW 4

5. I NTRODUCTION 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

6. S ET -U P The magnetic field is at some point P The length element is The wire is carrying a steady current of I Please replace with fig. 30.1 6

7. O BSERVATIONS The vector is perpendicular to both and to the unit vector directed from toward P The magnitude of is inversely proportional to r 2, where r is the distance from to P The magnitude of is proportional to the current and to the magnitude ds of the length element The magnitude of is proportional to sin  where  is the angle between the vectors and 7

8. The observations are summarized in the mathematical equation called the Biot-Savart law : 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 E QUATION 8

9. P ERMEABILITY OF F REE S PACE The constant  o is called the permeability of free space  o = 4  x 10 -7 T. m / A 9

10. T OTAL M AGNETIC F IELD is 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 The integral is over the entire current distribution 10

11. F INAL N OTES The law is also valid for a current consisting of charges flowing through space represents 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

12. FOR A L ONG, S TRAIGHT C ONDUCTOR The thin, straight wire is carrying a constant current Integrating over all the current elements gives 12

13. FOR A L ONG, S TRAIGHT C ONDUCTOR, S PECIAL C ASE If the conductor is an infinitely long, straight wire,    =  /2 and    = -  /2 The field becomes 13

14. FOR A C URVED W IRE S EGMENT Find the field at point O due to the wire segment I and R are constants  will be in radians 14

15. S PECIAL C ASES 15

16. FOR A C IRCULAR L OOP OF W IRE Consider the previous result, with a full circle  = 2  This is the field at the center of the loop 16

17. FOR A C IRCULAR C URRENT L OOP The loop has a radius of R and carries a steady current of I Find the field at point P 17

18. M AGNETIC F LUX 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

19. The magnetic field in this element is is a vector that is perpendicular to the surface has a magnitude equal to the area dA The magnetic flux F B is The unit of magnetic flux is T. m 2 = 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

20. G AUSS ’ L AW IN M AGNETISM 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: 20

21. A MPERE ’ S C IRCUITAL L AW 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

22. Where J is the total current density (in ampere per square metre, Am −2 ). 22

23. where J f 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 23

24. 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

25. 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 R 3 that contains S. Then, 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. ∫ C F ⋅ dr = ∫∫ curl F ⋅ dS S 25

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