*Gradient of a scalar function is a vector whose magnitude is equal to the maximum rate of increasing change of the scalar function per unit distance, and its direction is along the direction of maximum increase.

Gradient

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*Flux Density - amount of outward flux crossing a unit surface ds*Flux lines - field lines that comes out of the surface*divergence of a vector field - measure of the net outward flux per unit volume through a closed surface surrounding the unit volume

*transforms the volume integral of the divergence of a vector field into a surface integral of the field's flux through a closed surface surrounding the volume.

*If divergence is zero, it is solenoidal.

Divergence

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Rectangular Coordinate System:

Cylindrical Coordinate System:

Spherical Coordinate System:

Vector Identities Involving Curl:

(1) (2) (3)

Curl

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Laplacian Operator = divergence of the gradient of a scalar

Laplacian

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Electric Potential

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Charge and Current Distribution

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9.1 Force on a Moving Charge

9.2 Force on a Differential Current Element

List of Problems:

Hayt (Chapter 9 - Magnetic Forces, Materials and Inductance)

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Wednesday, May 20, 2015 7:45 AM

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Summary:

Introduction:

*Charge (stationary) produce electric fields. *Currents (non-time varying / steady) produce magnetic fields.* = Magnetic permeability*J = current density*D = E -> only applicable when the medium is linear and isotrophic, which allows us to treat the permittivity to be constant *B = H -> is constant on most dielectrics and metals except ferrormagnetic materials

5.1 Magnetic Forces and Torques

Electric Force is in the direction of the Electric Field.

Magnetic Force is perpendicular to the Magnetic Field.

Magnetostatics

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Electric Force acts whether the charge is moving or not.

Magnetic Force acts only when the charge is in motion.

Electric Force creates work. Magnetic Force doesn't create work.

Torque = force that tends the object to rotate

NIA = m = magnetic moment (A / m2) in the direction of the normal vector.d = perpendicular distancemoment arm = length of d

*When the thumb of the right hand is pointed along the direction of the torque, the four fingers indicate the direction that the torque is trying to rotate the body.*Direction of the perpendicular distance: from magnetic field to the current

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Questions about Magnetostatics

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Biot-Savart Law*Magnetic Flux Density (B) and magnetic field (H) are linearly related through magnetic permeability (): *Through his experiments on the deflection of compass needles by current-carrying wires, Hans Oersted established that currents induce magnetic fields that form close loops around the wires.*Jean Biot and Felix Savart arrived at an expression that relates the magnetic field H at any point in space to the current I that generates H. *Biot-Savart Law - the differential magnetic field dH generated by a steady current I flowing through a differential length dl is given by

To get the total magnetic field H due to a conductor of finite size:

where l is the line path along which I exists.

Magnetic Field due to Surface and Volume Current Distributions*The Biot-Savart law may also be expressed in terms of distributed current sources, such as the volume current density J, measured in (A/m2) or the surface current density J, measured in (A/m).

Magnetic Field of a Magnetic Dipole*Magnetic Dipole - A current loop with dimensions much smaller than the distance between the loop and observation.*Its pattern is similar to a permanent magnet.

Biot-Savart Law

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Thursday, May 21, 2015 8:08 PM

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Introduction:*Charges induce electric fields and currents induce magnetic fields.*Electromagnetic waves are produces when magnetic and electric fields couple, which is normally caused when charge and current sources were to vary with time.*Faraday's Law - time-varying magnetic field gives rise to an electric field*Ampere's Law - time-varying electric field gives rise to a magnetic field

6.1 Faraday's Law: *Oersted - showed that electricity and magnetism are closely connected.*He showed that a wire carrying an electric current exerts a force on a compass needle and that needle always turns so as to point the direction when the current is along the z-direction

*Faraday - "if a current can produce a magnetic field, then the converse should also be true!"*Faraday and Henry - "magnetic fields can produce an electric current in a closed loop, but only if the magnetic flux linking the surface area of the loop changes with time."*Galvanometer - sensitive instrument used in the 1800s to detect the flow of current in a circuit / predecessor of voltmeter and ammeter

Electromotive Induction can be done in three ways:

Type Magnetic Field Loop

1. Transformer EMF Time-varying Stationary

2. Motional EMF Static Moving loop with time-varying area

3. Combined Time-varying Moving loop

Chapter 6 - Maxwell's Equations for Time Varying Fields

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6.2 Stationary Loop in a Time Varying Magnetic Field

*R1 is usually very small and it may be ignored.*Lenz Law - the current in the loop is always in such a direction as to oppose the change of magnetic flux (t) that produced it

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Questions on Faraday's Law

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