1 Magnetostatics The Basics

2 Stationary charge: Stationary charge: v q = 0 E  0B = 0 Moving charge:Moving charge: v q  0 and v q = constant E  0B  0 Accelerating charge:Accelerating charge: v q  0 and a q  0 E  0B  0 Radiating field A uniformly moving charge produces an electric and magnetic field. A stationary charge produces an electric field only. A accelerating charge produces an electric and magnetic field and a radiating electromagnetic field.

3 Units and definitions Magnetic field strength Magnetic field vector Magnetic induction Magnetic flux density Tesla Gauss SI unit Weber

4 Permeability Permeability of free space Relative permeability for a medium Permeability of the medium Exact Exact constant

5 ferromagnetic

6 What happens when we cycle the applied magnetic field.

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9 Magnetostatics 1 st Postulate

POSTULATE 1 FOR THE MAGNETIC FIELD A current element immersed in a magnetic field will experience a force given by: Units of Newtons {N}

11 POSTULATE 1 FOR THE MAGNETIC FIELD POSTULATE 1 FOR THE MAGNETIC FIELD A current element experiences a force which is at right angles to the plane formed by the current element and magnetic field direction magnitude:

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13 postulate 1 for the magnetic field postulate 1 for the magnetic field Consider a straight segment Net force on the segment Right hand rule for direction

14 postulate 1 for the magnetic field postulate 1 for the magnetic field Magnetic force on a moving charge Current density: Current through cross section dA: Volume charge density Where is an element of volume enclosing charge q

15 postulate 1 for the magnetic field postulate 1 for the magnetic field Magnetic force on a moving charge Modify force equation: Net force on charge q X

16 postulate 1 for the magnetic field postulate 1 for the magnetic field Magnetic force on a moving charge Lorentz force Often used to define the magnetic field. Force, charge, and charge velocity are measurable.

17 Magnetic force on a moving charge

 Force at right angle to v and B vectors  Force proportional to v  Can do no work on a charge  Force along electric field lines  Force independent of v  Can do work 18 0 always

22 Hall effect

23 Hall effect

24 q 3-D view of block When a conductor that carries a current is placed in a uniform magnetic field, an electrostatic field appears whose direction is perpendicular both to the magnetic field and to the current. The electric field here is known as the Hall field and reaches equilibrium in the order of s. The electric field is characterized also by the Hall voltage across the faces of the conductor.

25 q x y z 3-D view of block

26 q x y z Top view of block Look onto this surface from above Negative charge build up on this surface Positive charge build up on this surface

27 q Top view of block Accumulation of charge continues until induced electric force equals magnetic force Voltage difference across charge distribution

28 q 3-D view of block w t Current density Current with N density of carriers in the material. get v

29 q 3-D view of block w t Velocity of the moving charge Magnetic force Simplified magnetic force on the charge moving at velocity v

30 q Top view of block Accumulation of charge continues until induced electric force equals magnetic force Voltage difference across charge distribution with w and

31 q In the steady state w then

32 Magnetostatics 2 nd Postulate

33 POSTULATE 2 FOR THE MAGNETIC FIELD POSTULATE 2 FOR THE MAGNETIC FIELD A current element produces a magnetic field which at a distance R is given by: Units of {T, G, Wb/m 2 }

34 POSTULATE 2 FOR THE MAGNETIC FIELD POSTULATE 2 FOR THE MAGNETIC FIELD Postulate 2 implies that the magnetic field is everywhere normal to the element of length

35 Magnitude of the magnetic field Magnitude of the magnetic field Similar to: Conceptually similar to a magnetic charge. Magnetic charges have not yet been found.

36 For a closed path made up of current elements For a closed path made up of current elements Biot-Savard Law

37 Magnetic field produced by extended conductor Magnetic field produced by extended conductor

38 Magnetic field produced by single moving charge conductor Magnetic field produced by single moving charge conductor

39 POSTULATE 1 and 2 FOR THE MAGNETIC FIELD POSTULATE 1 and 2 FOR THE MAGNETIC FIELD Magnetic field lines are continuous and close on themselves. There are no magnetic charges for the lines to start or end on. Magnetic forces and magnetic fields are at 90 degrees to their sources.

 From postulate 2: A moving charge produces a magnetic field.  From postulate 1: A magnetic field produces a force on a moving charge.  Is it possible then that a moving charge generate a magnetic force on a second moving charge? 40

Is it possible then that a moving charge generate a magnetic force on a second moving charge?  Answer “YES” 41

42 Recall for a moving charge that the following substitution is possible: THEN BECOMES

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44 The magnetic force on these two moving charges can be obtained from the elctric force and the velocity of the two charges. Concepts of relativity come into play. maximum