 # Maxwell’s equations

The conversion of electrical energy in machines is carried out using electromagnetic fields, the mathematical description of which in the most general form corresponds to Maxwell’s equations. The first equation establishes the relationship between electric current and magnetic field strength,Maxwell’s equations where and are the vectors of the magnetic field strength and elementary displacement along the closed loop, and / H dl i dq dt = is the total current passing through the surface bounded by the integration contour, and including conduction and bias currents.

This equation quantitatively characterizes the magnetic field that occurs when electric charges move, as well as when the electric field changes, because its right side includes the total current, including the bias current in vacuum. In the case of slowly changing fields, the bias current on the right side of equation (1.17) can be neglected. Then in this equation only the conduction current will be taken into account, and it will correspond to a law called the law of total current. It can be formulated as follows: the linear integral of the magnetic field vector taken over a closed circuit is equal to the total electric current passing through the surface bounded by this current.

The current on the right side of equation (1.17) is called the magnetizing force (NS). It is measured in units of current – amperes. The unit of measurement of the magnetic field is amperes per meter [A / m]. If the integration loop can be divided into n sections, within which the field is uniform and the direction of the path in each section is chosen to coincide with the direction of the intensity vector, then the integral in (1.17) can be replaced by the sum of the products of the stresses by the length of the sections: If, in this case, the integration circuit passes along the axis of the coil with the number of turns w and current I, then on the right side, instead of the sum of currents, there will be the product Iw.

The connection of the magnetic field with the electric was discovered by Faraday and is called the law of electromagnetic induction. Maxwell generalized this law to the case of an arbitrary medium, and in this form he received the name of Maxwell’s second law. As applied to closed loops, the law of electromagnetic induction in the formulations of Faraday and Maxwell is identical. In accordance with Faraday’s law, an electromotive force acting along a circuit and numerically equal to the linear integral of the electric field vector taken along the circuit is equal to the rate of change of the magnetic flux flow taken with the opposite sign Expression (1.19) is valid for any closed loop, not necessarily formed by a conductor. In the general case, this circuit can also be an imaginary circuit located entirely in the dielectric or partially in the conducting medium and partially in the dielectric. In all cases without exception, when the magnetic flux penetrating a surface bounded by any contour changes in time, an electromotive force arises in it, determined by this equation.

Equation (1.19) indicates that when the magnetic field changes in time in the same space, an electric field arises. Thus, the magnetic and electric fields are interconnected, and are two sides of a single electromagnetic process called an electromagnetic field.

In the simplest case of a single circuit with electric current i, the magnetic flux interlocking with it is determined only by the loop current and is called the self-induction flux (flux linkage): The coefficient L is called the self-induction coefficient or circuit inductance.

Obviously, with a constant current, a change in the size and shape of the circuit, as well as environmental properties, will cause a change in flux linkage L Ψ. Change in flux linkage will lead to the emergence of EMF self-induction If over time the loop inductance does not change (const L =), then A negative sign in expressions (1.21) and (1.22) indicates that the induced EMF tends to induce a current directed in such a way as to prevent a change in the magnetic flux. This position is the Lenz rule and represents the principle of electromagnetic inertia.

## Reference Maxwell’s equations

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