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Electrostatics in Vacuum, in Conductors and in the Presence of Linear Dielectrics

Principle of Superposition. Electrostatics in Vacuum, in Conductors and in the Presence of Linear Dielectrics. Charges were at rest. Magnetostatics. Look into the Forces between the charges which are in motion ….the Types of Current distributions Continuity Equation

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Electrostatics in Vacuum, in Conductors and in the Presence of Linear Dielectrics

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  1. Principle of Superposition Electrostatics in Vacuum, in Conductors and in the Presence of Linear Dielectrics Charges were at rest

  2. Magnetostatics • Look into the Forces between the charges which are in motion • ….the Types of Current distributions • Continuity Equation • Magnetic Field of a Steady Current • The Divergence and Curl of B • Magnetic Vector Potential A

  3. The Forces between the charges which are in motion (a) Current in opposite direction (b) Current in same direction

  4. What we are encountering is:The Magnetic Force • Other Example is : Magnetic Compass ….. the Needle will point towards the direction of the local magnetic field …..for instance towards the Geographic North. What if it is in the vicinity of a current carrying wire??

  5. Current carrying wires v X B (Due to wire 1) F 1 2

  6. In the presence of an Electrostatic Field E …justified by the experiments as well…

  7. Problem of a charged particle in a Cyclotron: y v B R X x Q F z

  8. z E y B x …trajectory of a Charged particle in the presence of an Uniform Electric field which is at Right angles to a Magnetic Field.

  9. y Emcosωt x B z Practice Problem: Answer

  10. Magnetic Forces do not Work Homework Problem: Example 5.3

  11. ….the Types of Current distributions • Line Currents • Surface Currents • Volume Currents

  12. Line Currents

  13. Problem 5.4: Suppose that the magnetic field in some region has the form Find the force on a square loop, lying in the y-z plane, if it carries a current I, flowing counterclockwise, when you look down the x-axis.

  14. z 3 a I 2 y 4 x 1

  15. Surface Currents Flow K

  16. Problem 5.6 (a) A Phonograph record carries a uniform density of “static electricity” σ. If it rotates at angular velocity ω, what is the surface current density K at a distance rfrom the center? z ω 0 r y x

  17. Volume Currents Flow

  18. Problem 5.6(b) A uniformly charged solid sphere, of radius R and total charge Q, is centered at the origin and spinning at a constant velocity ωabout thezaxis. Find the current densityJ at any point (r,θ,Φ) within the sphere. z ω P θ r y Φ x

  19. Problem 5.5 A current I flows down a wire of radius a. (i) If it is uniformly distributed over the surface,what is the current density K ?(ii) If it is distributed in such a manner that the volume current density is inversely proportional to the distance from the axis, what is J? a z

  20. Problem: (a) A current I is uniformly distributed over a wire of circular cross-section, with radius a. Find the current density J. (b) Suppose the current density is proportional to the distance from the axis,J=ks. Find thetotalcurrent in the wire. a z

  21. The Continuity Equation The Arrows indicate charge leaving the volume V Q(t) …which is precisely the mathematical statement of local charge conservation.

  22. Magnetic Field of a Steady Current • Why Steady Current is required here and which type of magnetic fields do steady currents give rise to?? • What is the form of the continuity equation in this case? • ….and the “Biot-Savart Law”…

  23. The Biot-Savart Law: The magnetic field of a steady current is given by: I rs P

  24. P Ө1 rs Ө Ө2 s α I L/ Wire Segment I dL/ Long Wire Problem: Find the magnetic field a distance sfrom a long straight wire carrying a steady current I.

  25. Problem: Find the magnetic field a distance z above thecenterof acircular loop of radius R, which carries a steady current I. z rs R

  26. Ө2 P z Ө1 a Problem:5.11 Find the magnetic field at point P on the axis of a tightly wound solenoid (helical coil) consisting of n turns per unit length wrapped around a cylindrical tube of radius a and carrying current I.

  27. I I b a R P I I I R P I Problem: 5.9 Find the magnetic field at point P for each of the steady current configurations shown below:

  28. Problem:5.45 A semicircular wire carries a steady current I. Find the magnetic field at a point P on the other semicircle. P R Ө I

  29. Problem: 5.10(a) Find the force on the current carrying square loop due to a current carrying wire a I a s I

  30. Problem:5.46 The magnetic field on the axis of a circular current loop is far from uniform (it falls off sharply with increasing z). However, one can produce a more nearly uniform field by using two such loops a distance d apart. z R I z=0 d R I

  31. (a) Find the field B as a function of z, and show that ∂B/∂z is zero at the point midway between them (z=0). Now, if you pick d just right the second derivative of B will also vanish at the midpoint. (b) Determine d such that ∂2B/∂z2=0 at the midpoint, and find the resulting magnetic field at the center.

  32. This arrangement is known as a Helmholtz coil; it’s a convenient way of producing relatively uniform fields in the laboratory. z R I z=0 d R I

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