Electromagnetic Theory EKT 241 Electromagnetic Theory Magnetostatic Anayet Karim

Chapter Objectives Magnetic force The total electromagnetic force, known as Lorentz force Biot–Savart law Gauss’s law for magnetism Ampere’s law Vector magnetic potential 3 different types of material Boundary between two different media Magnetic energy density

Chapter Outline 4-1) Magnetic Forces and Torques The Biot–Savart Law Magnetic Force between Two Parallel Conductors Maxwell’s Magnetostatic Equations Vector Magnetic Potential Magnetic Properties of Materials Magnetic Boundary Conditions Inductance Magnetic Energy 4-2) 4-3) 4-4) 4-5) 4-6) 4-7) 4-8) 4-9)

Today We Will Focus the Following Topics Magnetic Forces Apart between Fe & Fm Magnetic Torques Few Examples

4-1 Magnetic Forces and Torques When charged particle moving with a velocity u, magnetic force Fm is produced. where B = magnetic flux density (C.m/s or Tesla T) When charged particle has E (electric field) and B (Magnetic field), total electromagnetic force is

Apart Fe & Fm Fe always in the direction of electric field Fm always perpendicular to magnetic field Fe acts on a charged particles, which is static Fm acts on a charged particles, which is moving or in motion E & D (Electric field & Density) H & B (magnetic field & Density) Fe=qE for “Force on charge q) Fm = quXB “Force on charge q) Stationary charges (Electrostatics) & Steady currents (Magnetostatics)

4-1.1 Magnetic Force on a Current-Carrying Conductor For closed circuit of contour C carrying I , total magnetic force Fm is Fm is zero for a closed circuit. On a line segment, it is proportional to the vector between the end point.

4-1.1 Magnetic Force on a Current-Carrying Conductor The total magnetic force Fm on any closed current loop in a uniform magnetic field is zero

Example 4.1 Force on a Semicircular Conductor The semicircular conductor shown lies in the x–y plane and carries a current I . The closed circuit is exposed to a uniform magnetic field . Determine (a) the magnetic force F1 on the straight section of the wire and (b) the force F2 on the curved section. a) b) Solution

4-1.2 Magnetic Torque on a Current-Carrying Loop Applied force vector F and distance vector d are used to generate a torque T T = d× F (N·m) Rotation direction is governed by right-hand rule. Magnetic Field in the Plane of the Loop F1 and F3 generates a torque in clockwise direction.

4-1.2 Magnetic Torque on a Current-Carrying Loop Magnetic Field Perpendicular to the Axis of a Loop When loop consists of N turns, the total torque is The vector m with normal vector is expressed as T can be written as

What we learn today Electric force Fe Magnetic force Fm Electromagnetic force F Lorentz force Apart between Fe & Fm (5 types) Examples are very Important

Test Yourself What is Force Apart between Force & Torque What is 2D axis What is 3D axis

Q & A Feel Free to ASK ME Next Class would be various laws- Do some study in advance

THANK YOU TERIMA KASIH

4-2 The Biot–Savart Law Biot–Savart law states that where dH = differential magnetic field dl = differential length To determine the total H we have

4-2.1 Magnetic Field due to Surface and Volume Current Distributions Biot–Savart law may be expressed in terms of distributed current sources.

Example 4.3 Magnetic Field of a Pie-Shaped Loop Determine the magnetic field at the apex O of the pie-shaped loop as shown. Ignore the contributions to the field due to the current in the small arcs near O. For segment AC, Consequently, Solution

4-2.2 Magnetic Field of a Magnetic Dipole To find H in a spherical coordinate system, we have where R’ >> a

4-3 Magnetic Force between Two Parallel Conductors Force per unit length on parallel current-carrying conductors is where F’1 = -F’2 (attract each other with equal force)

4-4 Maxwell’s Magnetostatic Equation There are 2 important properties: Gauss’s and Ampere’s Law. Gauss’s law for magnetism states that Net electric magnetic flux through a closed surface is zero. 4-4.1 Gauss’s Law for Magnetism

4-4.2 Ampere’s Law Ampere’s law states that The directional path of current C follows the right-hand rule.

Example 4.6 Magnetic Field inside a Toroidal Coil A toroidal coil (also called a torus or toroid) is a doughnut-shaped structure (called its core) with closely spaced turns of wire wrapped around it as shown. For a toroid with N turns carrying a current I , determine the magnetic field H in each of the following three regions: r < a, a < r < b, andr > b, all in the azimuthal plane of the toroid.

Solution 4.6 Magnetic Field inside a Toroidal Coil H = 0 for r < a as no current is flowing through the surface of the contour H = 0 for r > b, as equal number of current coils cross the surface in both directions. Application of Ampere’s law then gives Hence, H = NI/(2πr) and

4-5 Vector Magnetic Potential For any vector of vector magnetic potential A, We are able to derive . Vector Poisson’s equation is given as where

4-6 Magnetic Properties of Materials Magnetic behavior is due to the interaction of dipole and field. 3 types of magnetic materials: diamagnetic, paramagnetic, and ferromagnetic. Electron generates around the nucleus and spins about its own axis. 4-6.1 Orbital and Spin Magnetic Moments

4-6.2 Magnetic Permeability Magnetization vector M is defined as where = magnetic susceptibility (dimensionless) Magnetic permeability is defined as and relative permeability is defined as

4-6.3 Magnetic Hysteresis of Ferromagnetic Materials Ferromagnetic materials is described by magnetized domains. Properties of magnetic materials are shown below.

4-6.3 Magnetic Hysteresis of Ferromagnetic Materials Comparison of hysteresis curves for (a) a hard and (b) a soft ferromagnetic material is shown.

4-7 Magnetic Boundary Conditions For 2 different media when applying Gauss’s law, we have Boundary condition for H is . Vector defined by the right-hand rule is At interface between media with finite conductivities, Js = 0 and .

4-8 Inductance An inductor is a magnetic capacitor. An example is a solenoid as shown below.

4-8.1 Magnetic Field in a Solenoid For a cross section of solenoid, When l >a, θ1≈−90° and θ≈90°,

4-8.2 Self-Inductance Magnetic flux is given by To compute the inductance we need area S.

4-8.2 Self-Inductance The self-inductance L of conducting structure is defined as where = total magnetic flux (magnetic flux linkage) For a solenoid, For two-conductor configurations,

4-8.3 Mutual Inductance Magnetic field lines are generated by I1 and S2 of loop 2. The mutual inductance is Transformer uses torodial coil with 2 windings.

4-9 Magnetic Energy Total energy (Joules) expended in building up the current in inductor is Some of the energy is stored in the inductor, called magnetic energy, Wm. The magnetic energy density wm is defined as

Example 4.9 Magnetic Energy in a Coaxial Cable Derive an expression for the magnetic energy stored in a coaxial cable of length l and inner and outer radii a and b. The insulation material has permeability µ. The magnitude of the magnetic field is Magnetic energy stored in the coaxial cable is given by Solution