ELEC 3105 Basic EM and Power Engineering Magnetostatics The Basics

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

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

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

Ferromagnetic materials WHY? What happens when we cycle the applied magnetic field.

Permittivity and Permeability

ELEC 3105 Basic EM and Power Engineering Magnetostatics 1st Postulate

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

Magnetostatics 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:

Magnetostatics vs Electrostatics

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

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

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

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

Magnetic force on a moving charge

Magnetic Electric 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 0 always

Electric AND Magnetic

ELEC 3105 Basic EM and Power Engineering Hall effect

Hall effect

Hall Effect 3-D view of block q 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 10-14 s. The electric field is characterized also by the Hall voltage across the faces of the conductor.

Hall Effect 3-D view of block z q y x

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

Hall Effect Top view of block Accumulation of charge continues until induced electric force equals magnetic force. - - - - - - - - - - - - q + + + + + + + + + + Voltage difference across charge distribution

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

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

Hall Effect Accumulation of charge continues until induced electric force equals magnetic force. Top view of block - - - - - - - - - - - - q w with + + + + + + + + + + and Voltage difference across charge distribution

Hall Effect In the steady state - - - - - - - - - - - - then q w - - - - - - - - - - - - then q w + + + + + + + + + +

ELEC 3105 Basic EM and Power Engineering Magnetostatics 2nd Postulate

Magnetostatics 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/m2}

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

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

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

Magnetostatics Magnetic field produced by extended conductor

Magnetostatics Magnetic field produced by single moving charge conductor

Magnetostatics 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.

Consequence of postulates 1 and 2 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?

Consequence of postulates 1 and 2 Is it possible then that a moving charge generate a magnetic force on a second moving charge? Answer “YES”

Consequence of postulates 1 and 2 Recall for a moving charge that the following substitution is possible: THEN BECOMES

Compare magnitude of magnetic and electric force between two moving charges

Compare magnitude of magnetic and electric force between two moving charges 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

Magnetic induction charging

Induction Stove