Electromagnetic I EELE 3331

Lecture XMagnetostatic Forces

Dr. Mohamed OudaElectrical Engineering DepartmentIslamic University of Gaza

Magnetic Force on a Moving Charge

Electric charges moving in a magnetic field experience a force.

Given Q moving with velocity u in B, the vector magnetic force Fm on the charge is given by

Note that the force is normal to the plane containing the velocity vector and the magnetic flux density vector.

Also note that the force is zero if the charge is stationary (u=0).

Example

Determine the vector magnetic force on a point charge +Q moving at a uniform velocity u =uo ay

in a uniform magnetic flux density B =Bo az .

Lorentz force equation

a charge moving in an electric and a magnetic fields

The Lorentz force can also be written in terms ofNewton’s law such that

where m is the mass of the charged particle.

Magnetic Force on Current

Nn

The equivalence of the moving point charge and the differential length of line current yields the equivalent magnetic force equation.

,

The overall force on a line, surface and volume current is found by integrating over the current distribution.

Example (Force between line currents)

Determine the force/unit length on a line current I1 due to magnetic flux produced by I2 as shown.

Torque on a Current Loop

Change current directions around a closed current loop, Varying direction of the magnetic forces on different portions of the loop.

Lorentz force equation shows that the net force on a simple circular or rectangular loop is a torque which forces the loop to align its magnetic moment with the applied magnetic field.

For a rectangular current loop lies in the x-y plane and carries a DC current I. The loop lies in a uniform magnetic flux density B given by

T on the loop is defined in terms of the force magnitude (l By I2 ), the torque moment arm distance (l1 /2), and the torque direction (defined by the right hand rule):

The vector torque can be written compactly by defining the magnetic moment (m) of the loop

Magnetization

Certain materials can be magnetized under the

influence of an applied magnetic field

The poles of the bar magnet can be represented as equivalent magnetic charges separated by a distance l.

The magnetic flux density produced by the magnetic dipole is equivalent to the electric field produced by the electric dipole.

A current loop and a solenoid produce the same B as the bar magnet at large distances (in the far field) if the magnetic moments of these three devices are equivalent.

Magnetic Materials

Diamagnetic Materials

relative permeability of just under one (a small negative magnetic susceptibility)

the magnetic moments due to electron orbits and electron spin are very nearly equal and opposite such that they cancel each other. Thus, the response to an applied magnetic field is a slight magnetic field in the opposite direction.

Superconductors exhibit perfect diamagnetism

( ) at temperatures near absolute zero such that magnetic fields cannot exist inside these materials.

Paramagnetic Materials

Relative permeability is just greater than one

The magnetic moments due to electron orbit and spin are unequal, resulting in a small positive magnetic susceptibility.

Magnetization is not significant

Both diamagnetic and paramagnetic materials are typically linear media.

Ferromagnetic materials

Relative permeability much greater than one

Always nonlinear.

As such, these materials cannot be described by a single value of relative permeability.

If a single number is given for the relative permeability of any ferromagnetic material, this number represents an average value of µr .

lose their ferromagnetic properties at very high temperatures (above a temperature known as the Curie temperature).

hysteresis loop

Magnetic Boundary Conditions

Given a surface current on the interface, the tangential magnetic field components on either side of the interface point in opposite directions.

Normal Magnetic Flux Density

Inductors and Inductance

An inductor is an energy storage device that stores energy in an magnetic field.

flux linkage

The flux linkage of an inductor defines the total magnetic flux that links the current.

If the magnetic flux produced by a given current links that same current, the resulting inductance is defined as a self inductance.

If the magnetic flux produced by a given current links the current in another circuit, the resulting inductance is defined as a mutual inductance.

Example (Transformer / self and mutual inductance)

Mutual Inductance Calculations

The mutual inductance between two distinct circuits can be determined by assuming a current in one circuit and determining the flux linkage to the opposite circuit.

Example

M

Given loop 2 is much smaller than loop 1, then flux produced by loop 1 is nearly uniform over the area of loop 2 and is approximately equal to that at the loop center. The flux produced by loop 1 can also be assumed to be approximately normal to loop 2 over its area. Thus, the total flux produced by loop 1 linking loop 2 is approximately

Internal and External Inductance

In general, a current carrying conductor has magnetic flux internal and external to the conductor.

Thus, the magnetic flux inside the conductor can link portions of the conductor current which produces a component of inductance designated as internal inductance.

The magnetic flux outside the conductor that links the conductor current is designated as external inductance.

The most efficient technique in determining the internal and external components of inductance is the energy method.

The energy method for determining inductance is based on the total magnetic energy expressionfor an inductor given by

Example

Determine the internal inductance for a cylindrical conductor of radius a carrying a uniform current density J.

The magnetic flux density internal to the conductor is given by

The internal inductance per unit length of a non-magnetic ) is conductor

Example

Determine the external inductance for a coaxial transmission line assuming uniform current densities in both the inner conductor (radius=a) and the outer conductor (inner radius=b).

Example