Chapter 8. Steady-state magnetic field EMLAB

B (Magnetic flux density), H (Magnetic field) Magnetic field is generated by moving charges, i.e. current. If current changes with time, electric field is generated by time varying magnetic field. In chapter 8, we consider only steady state current. In this case, steady magnetic fields are generated and we need consider magnetic field only. If a charge moves in a region where magnetic flux density is non-zero, it experiences a force due to the field which is called Lorentz force. The force exerted on a moving charge is due to B (Magnetic flux density). B can be obtained from a magnet or current flowing coil. B due to current flowing coil only is defined to be H (magnetic field). B due to a permanent magnet is represented by M (magnetization). EMLAB

Biot-Savart law This law is discovered by Biot and Savart. It enables us to predict magnetic field due to a current segment. This law is experimentally known. It is the counterpart of Coulomb’s law for electric field. Current segment Direction of H-field EMLAB

Biot-Savart law : integral form Line current surface current Volume current EMLAB

Magnetic field due to an infinitely long line current An infinitely long straight current flowing in the z-axis. odd function EMLAB

Magnetic field due to a finitely long current filament 기함수 EMLAB

Magnetic field due to a loop current Magnetic field on the z-axis can only be found due to its simple shape. If the receiver’s position is located on the off-axis region, the integral can be evaluated. EMLAB

Calculation of H of a solenoid Surface current density If a copper wire is wound around a cylinder N times in the length d and current I is flowing through it, it can be approximated by a surface current along  direction with a magnitude K = NI/d. EMLAB

Calculation of integral 1 (1) If r is outside V, the integral becomes zero in that Laplacian ϕ becomes zero. (2) If r is inside V, the integral can be changed into a surface integral over a enclosing sphere, which has non-zero value. r 근방에서의 적분을 계산하기 위해서 r 을 중심으로 하고 반지름이  인 구면에서의 면 적분을 하면 결과는 1이 나와서 앞 장의 결과가 나온다. EMLAB

Ampere’s law Ampere law facilitates calculation of mangetic field like the Gauss law for electric field.. Unlike Gauss’ law, Ampere’s law is related to line integrals. Ampere’s law is discovered experimentally and states that a line integral over a closed path is equal to a current flowing through the closed loop. In the left figure, line integrals of H along path a and b is equal to I because the paths enclose current I completely. But the integral along path c is not equal to I because it does not encloses completely the current I. EMLAB

Example- Coaxial cable The direction of magnetic fields can be found from right hand rule. The currents flowing through the inner conductor and outer sheath should have the same magnitude with different polarity to minimize the magnetic flux leakage EMLAB

Example : Surface current The direction of magnetic field con be conjectured from the right hand rule. EMLAB

Example : Solenoid The direction of magnetic field con be conjectured from the right hand rule. If the length of the solenoid becomes infinite, H field outside becomes 0. EMLAB

Example : Torus EMLAB

Example problem 8.18 A wire of 3-mm radius is made up of an inner material (0 < ρ < 2 mm) for which σ = 107 S/m, and an outer material (2mm < ρ < 3mm) for which σ = 4×107S/m. If the wire carries a total current of 100 mA dc, determine H everywhere as a function of ρ. 2 100mA 1 2mm EMLAB

Example problem 8.9 EMLAB

Example problem 8.6 EMLAB