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# Chapter 29 Sources of Magnetic Fields

## Magnetic Field due to Moving Charges

A stationary charge creates and interacts with the electric field, while a moving charge creates a magnetic field. Consider a point charge moving…

## Biot-Savart Law

The Biot-Savart law gives the magnitude and direction of the magnetic field produced by a current. This empirical law was named in honor of two…

## Biot-Savart Law: Problem-Solving

The magnitude and direction of a magnetic field created by a steady current can be calculated using the Biot-Savart law. Consider a mobile phone…

## Magnetic Field Due To A Thin Straight Wire

Consider an infinitely long straight wire carrying a current I. The magnetic field at point P at a distance a from the origin can be calculated using…

## Magnetic Field Due to Two Straight Wires

Consider two parallel straight wires carrying a current of 10 A and 20 A in the same direction and separated by a distance of 20 cm. Calculate the…

## Magnetic Force Between Two Parallel Currents

Two long, straight, and parallel current-carrying conductors exert a force of equal magnitude on one another. The direction of the force depends on…

## Magnetic Field Of A Current Loop

Consider a circular loop with a radius a, that carries a current I. The magnetic field due to the current at an arbitrary point P along the axis of…

## Divergence and Curl of Magnetic Field

The magnetic field due to a volume current distribution given by the Biot–Savart Law can be expressed as follows: To evaluate the divergence…

## Ampere's Law

A fundamental property of a static magnetic field is that it is not conservative, unlike an electrostatic field. Instead, there is a relationship…

## Ampere's Law: Problem-Solving

Ampere's law states that for any closed looped path, the line integral of the magnetic field along the path equals the vacuum permeability times…

## Solenoids

A solenoid is a conducting wire coated with an insulating material, wound tightly in the form of a helical coil. The magnetic field for a solenoid is…

## Magnetic Field of a Solenoid

A solenoid is a conducting wire coated with an insulating material, wound tightly in the form of a helical coil. The magnetic field due to a solenoid…

## Toroids

A toroid is a closely wound donut-shaped coil constructed using a single  conducting wire. In general, it is assumed that a toriod consists of…

## Magnetic Vector Potential

In electrostatics, the electric field can be written as the negative gradient of the potential. In magnetostatics, the zero divergence of the…

## Potential Due to a Magnetized Object

Magnetic dipoles in magnetic materials are aligned when placed under an external magnetic field. For paramagnets and ferromagnets, dipole alignment…

## Magnetostatic Boundary Conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular…

## Ampere's Law in Matter

The total current density in magnetized material is the sum of the free and bound current densities. The free current arises due to the motion of…

## Magnetic Susceptibility and Permeability

In linear magnetic materials, like paramagnets and diamagnets, magnetization is proportional to the magnetic field intensity. The constant of…

## Magnetic Moment of an Electron

Electrons revolving around a nucleus are analogous to a circular current carrying loop. This current produces a magnetic dipole moment proportional…

## Diamagnetism

Materials consisting of paired electrons have zero net magnetic moments. However, when these materials are placed under an external magnetic field,…

## Paramagnetism

Paramagnets are materials with unpaired electrons that possess a finite magnetic moment. In the absence of a magnetic field, these moments are…

## Ferromagnetism

Materials like iron, nickel, and cobalt consist of magnetic domains, within which the magnetic dipoles are arranged parallel to each other. The…

## Design of a Cyclic Pressure Bioreactor for the Ex Vivo Study of Aortic Heart Valves

The aortic valve, located between the left ventricle and the aorta, allows for unidirectional blood flow, preventing backflow into the ventricle.…

## Exploring the Radical Nature of a Carbon Surface by Electron Paramagnetic Resonance and a Calibrated Gas Flow

While the first Electron Paramagnetic Resonance (EPR) studies regarding the effects of oxidation on the structure and stability of carbon radicals…

## Use of Electron Paramagnetic Resonance in Biological Samples at Ambient Temperature and 77 K

The accurate and specific detection of reactive oxygen species (ROS) in different cellular and tissue compartments is essential to the study of…

## Quantifying the Relative Thickness of Conductive Ferromagnetic Materials Using Detector Coil-Based Pulsed Eddy Current Sensors

Thickness quantification of conductive ferromagnetic materials by means of non-destructive evaluation (NDE) is a crucial component of structural…

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