If there is no J current, the Struve function coefficient ( ) will be zero. Due to the symmetry of the system, the potential does not depend on the angle θ. Here, and are Bessel functions of first and second kind, respectively, and L is the modified Struve function of first kind 15. The above equation is a Modified Bessel equation and its response is also a Bessel function of second kind. In order for (21) to cover all the defined points (region Ω), each statement in parentheses in (21) must have a constant value. So by substituting (20) into (13), we get: Thus, the Laplacian of a function ψ is written in the cylindrical coordinate system as: The solution of (14) in a cylindrical coordinate system can be determined using the method of separating variables. Where is the vector position of the point where we calculate the potential A and is the vector position at the current position. If the boundary of the region is assumed infinite and A (∞) → 0, the solution is as follows 12: In this system, each equation is a Poisson equation. If the region Ω is simply connected area of (12) then: So, by replacing the vector potential in (3) we have: The gauge condition can be used to assume that the vector potential is unique, with 10: The divergence of the curl of each vector is null:Īccording to (4) and (9), the vector B can be expended as the curl of a vector potential 10. The Curl of the gradient of a numerical field is zero (the existence of V and its derivatives is implicitly assumed in all points): Hence 13:Īt this step, two identities can be useful to further derive the equations 13: The Helmholtz theorem suggests that, via the operator, any vector F can be expressed as a sum of the gradient of a scalar function V and the curl of a vector function A. In this study, we calculated the magnetic force between a coil with high permeability core and a coaxial permanent magnet using the Lorentz's theory of potential and force. They have shown that the shell method is an efficient and faster way to design many magnetic stimuli 9. calculated the force between a coil with an air core and a coaxial permanent magnet with four different methods (shell method, finite element method, and two integral methods) and compared them. They used mutual inductance between two equivalent coils to calculate the force between the two magnets 8. Then, they calculated the magnetic field using the potential theory. They used the equivalent of a magnet with a current-carrying coil to calculate the magnetic force. calculated the force between two permanent coaxial magnets. In design optimization, the effect of geometric parameters such as length and radius of coils and magnet was investigated 7. In this design, they calculated the total magnetic field using the Biot-Savart law. designed and optimized an electromagnetic soft actuator based on a coil with a permanent magnet core. The calculations in this study were performed by equating its electric current by fictitious magnetization 6. Ciric calculated the magnetic field within a transformer using the magnetic scalar potential method. The space of the coiled coil was considered to be completely homogeneous in these calculations, a constant permeability was considered for the whole space 2. calculated the magnetic field from a bobbin and coil using the magnetic field vector potential method. calculated the force between two coaxial coils using the filament method as an efficient numerical method 5. calculated the magnetic force between a thin cylindrical coil and a coaxial cylindrical magnet by deriving the mutual inductance between them 4, while A. Derived directly from the well-known Maxwell's equations 2, the equivalent magnetic charge model and equivalent magnetizing current are widely used to obtain, either analytically or numerically, the magnetic field and magnetic force of permanent magnets 3. Generation of controlled magnetic fields has many applications, including spacecraft, magnetic resonance imaging, bioelectromagnetic research, and nondestructive testing (NDT). Some devices, such as sensors, actuators, and speakers, are built with magnets, coils, or both 1.Įlectromagnetic coils are very important in modern medical and electrical sensors and systems. However, some devices, such as magnetic couplings, are made of permanent cylindrical magnets, while others, such as transformers, are made of coils. Magnetic devices used in electrical engineering require magnets or coils.
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