* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Study Guide
Magnetometer wikipedia , lookup
Magnetotactic bacteria wikipedia , lookup
Superconducting magnet wikipedia , lookup
Electric charge wikipedia , lookup
Giant magnetoresistance wikipedia , lookup
Earth's magnetic field wikipedia , lookup
Electromagnetism wikipedia , lookup
Magnetic monopole wikipedia , lookup
History of electrochemistry wikipedia , lookup
Magnetoreception wikipedia , lookup
Electric dipole moment wikipedia , lookup
Electromagnet wikipedia , lookup
Electric machine wikipedia , lookup
Electromotive force wikipedia , lookup
Force between magnets wikipedia , lookup
Lorentz force wikipedia , lookup
Magnetochemistry wikipedia , lookup
Maxwell's equations wikipedia , lookup
Magnetohydrodynamics wikipedia , lookup
Magnetotellurics wikipedia , lookup
Mathematical descriptions of the electromagnetic field wikipedia , lookup
History of geomagnetism wikipedia , lookup
Electricity wikipedia , lookup
Multiferroics wikipedia , lookup
Ferromagnetism wikipedia , lookup
Intermediate Electricity and Magnetism Study Guide Chapter 2: Chapter 5: Coulumb’s Law Find the Electric Field at a point in space from a known charge configuration or distribution. The electric field is a function of the charge and the distance away from the charge. Biot Savart Find the Magnetic Field at a point in space from a known current or current distribution. The Magnetic field is a function the current and the distance away from the current (a cross product). Gauss’ Law Find the Electric Field using symmetry in the field due to a known charge configuration or distribution. To solve these problems we must choose a Gaussian surface where along the surface the Electric Field is constant due to symmetry. The Electric Field should be parallel or perpendicular to our Gaussian Surface, or the field could be zero. Ampere’s Law Find the Magnetic Field using symmetry (mirror rule) in the field due to a known current or current distribution. To solve these problems we must choose and Amperian Loop where along the loop the magnetic field is constant due to symmetry. The amperian loop should be perpendicular to the magnetic field; it is beneficial if the magnetic field is zero along a side of the loop. Electric Potential Find electric potential from a charge configuration or a distribution using Poisson’s and Laplace’s equations to break a scenario down to a scalar quantity. This makes it so that you do not have to deal with components making the problem simpler. From a known electric potential it is not too difficult to calculate the Electric field, or the charge distribution. Vector Potential Find the vector potential from a current distribution. The vector potential is handy because it is not too difficult to calculate the magnetic field or the current distribution from a known vector potential. Energy in a Field The energy in a field due to a charge distribution can be calculated using the Electric field, from the electric potential, or a charge configuration based on Coulumb’s Law. Capacitance The capacitance is a constant of proportionality, based purely on the geometry of the problem. Capacitance is useful for determining the Electric Potential, work or energy associated with the field. Chapter 3: Multipole Expansion When more detail in calculating the electric potential instead of an approximation it can be beneficial to use a multipole expansion. Multipole expansion of the vector potential When more detail in calculating the electric potential instead of an approximation it can be beneficial to use a multipole expansion. Chapter 4 Electric Dipoles in an Electric Field Electric dipole in an electric field becomes polarized. And because of the polarization it experiences a torque because electric dipoles will seek to align themselves with the electric field. Induced Dipole An induced dipole is caused by the separation of the electron and nucleus due to an electric field. An induced dipole will seek to anti align itself with the electric field. Magnetic Dipole The magnetic dipole moment of a loop is a function of the current and the area of the loop. Polarization Polarization is the dipole moment per unit volume. A polarized object emits an electric field. Magnetization If an object is placed in a magnetic field the object should become magnetized. When an object becomes magnetized it experiences a torque because it will seek to align parallel, or anitparallel with the magnetic field. When an object becomes magnetized it seeks to align itself somehow due to the magnetic field because of the magnetic dipole moment per unit volume (magnetization) that occurs. Magnetic dipole moment per unit volume emits a magnetic field. Bound Charges It can be easy to calculate the electric potential from a known polarization. The polarization determines the bound charges; the polarized bound charges emit their own electric field. Bound Current The volume and surface bound current can be found by knowing the magnetization. There is a large amount of bound current within an object. The bound currents emit their own magnetic field. Free Charges Free charges are the charges that are not a result of polarization. The combination of free and bound charges is the total charge and the enclosed charges will be the sum of both free and bound charges. Free charges might consist of electrons on a conductor or ions embedded in the dielectric material or whatever. Free Current Free current is the current that is not caused by the magnetic field. The free current is due to an electric potential difference. The free current distinction is a convenience because then we do not have to deal with the bound current. Electric Displacement D can be calculated from a modification of Gauss’ Law, but is extremely useful because then we do not need to worry about the bound charges, just the free charges. Knowing the electric displacement, and the electric field, will tell us the polarization per unit volume. Auxiliary Field H permits us to express Ampere’s law in terms of free current. H is extremely useful because it makes determine the magnetic field die to current much simpler. Calculating the Electric Field from the Electric Displacement Calculating the Electric field form the electric displacement is easy because it is a matter of dividing the electric displacement by the the permittivity of the material. The permittivity of the material is a function of the electric susceptibility. Calculating the Magnetic Field from the Auxillary Field Calculating the magnetic field from the auxillary field is easy because it is a matter of multiplying the auxillary field by the permeability of the material. The permeability of the material is a function of the magnetic susceptibility. EQUATIONS CHAPTER 2 coulombs law 1 𝑄 𝐸 = 4𝜋𝜀 𝑟 2 𝑟̂ 0 Gauss’s Law 𝑄𝑒𝑛𝑐 = ∫ 𝜌 𝑑𝜏 1 𝑑𝑎 = 𝜀 ∫ 𝜌 𝑑𝜏 ∮ 𝐸⃑ ∙ ⃑⃑⃑⃑ 0 1 ⃑⃑⃑⃑ = 𝑄𝑒𝑛𝑐 ∮ 𝐸⃑ ∙ 𝑑𝑎 𝜀 0 Stokes’ theorem ∇𝑥𝐸 =0 Potential 𝑟 𝑉(𝑟) ≡ − ∫𝑟𝑒𝑓 𝐸 ∙ 𝑑𝑙 𝐸 = −∇𝑉 1 𝑉(𝑟) = 4𝜋𝜀 ∫ 𝜌(𝑟 ′ ) 𝑟 0 Energy 𝑊 = 𝑞𝑉 1 𝑊 = 2 ∫ 𝜌𝑉𝑑𝜏 𝑊= 𝑊= 𝜀0 2 𝜀0 2 𝑑𝜏 ′ 𝑞 and 𝜌 = 𝑉 ∫ 𝐸 2 𝑑𝜏 (∫𝑉 𝐸 2 𝑑𝜏 + ∮𝑠 𝑉𝐸 ∙ 𝑑𝑎) Capacitance 𝑄 𝐶=𝑉 Force 𝑓 = 𝜌𝐸⃑ 𝐹 = ∫ 𝑓 𝑑𝜏 CHAPTER 3 Multipoles 1 1 ′ 𝑛 ′ 𝑉(𝑟) = 4𝜋𝜀 ∑∞ 𝑛=0 𝑟 (𝑛+1) ∫(𝑟 ) 𝑃𝑛 (𝑐𝑜𝑠𝜃′)𝜌(𝑟 )𝑑𝜏′ 0 1 𝑉𝑚𝑜𝑛 (𝑟) = 4𝜋𝜀 𝑄 0 𝑟 𝜌 𝐸𝑑𝑖𝑝 (𝑟, 𝜃) = 4𝜋𝜀 0𝑟 3 (2𝑐𝑜𝑠𝜃𝑟̂ + 𝑠𝑖𝑛𝜃𝜃̂) CHAPTER 4 Torque 𝜌 = 𝛼𝐸 𝑁=𝑝𝑥𝐸 𝐹 = (𝑝 ∙ ∇)𝐸 bound charges 𝜎𝑏 = 𝑃 ∙ 𝑛̂ 𝜌𝑏 = −∇ ∙ 𝑃 1 𝑉(𝑟) = 4𝜋𝜀 ∮𝑠 𝜎𝑏 0 𝓇 1 𝑑𝑎′ + 4𝜋𝜀 ∫𝑉 0 𝜌𝑏 𝓇 𝑑𝜏′ 𝑄𝑡𝑜𝑡 = ∮ 𝜎𝑏 𝑑𝑎 + ∫𝑉 𝜌𝑏 𝑑𝜏 Electric Displacement 𝐷 ≡ 𝜀0 𝐸 + 𝑃 ∇ ∙ 𝐷 = 𝜌𝑓 ∮ 𝐷 ∙ 𝑑𝑎 = 𝑄𝑓𝑒𝑛𝑐 Linear Dielectrics 𝑃 = 𝜀0 𝜒𝑒 𝐸 𝐷 = 𝜀𝐸 𝜀 = 𝜀0 (1 + 𝜒𝑒 ) 1 𝑊 = 2 ∫ 𝐷 ∙ 𝐸 𝑑𝜏 CHAPTER 5 Forces 𝐹𝑚𝑎𝑔 = 𝑄(𝑣 x 𝐵) 𝐹 = 𝑄[𝐸 + (𝑣 x 𝐵)] 𝐹𝑚𝑎𝑔 = ∫ 𝐼(𝑑𝑙 x 𝐵) bound currents 𝐾 = 𝜎𝑣 𝐽 = 𝜌𝑣 𝐼 = ∫ 𝐽 ∙ 𝑑𝑎 surface 𝐼 = ∫𝑉 (∇ ∙ 𝐽)𝑑𝜏 𝐼 𝐽=𝐴 𝐽 = 𝜌𝑣 Biot-Savart 𝜇 𝐼 x 𝓇̂ 𝜇 (𝑑𝑙′ x 𝓇̂) 𝐵(𝑟) = 4𝜋0 ∫ 𝓇 2 𝑑𝑙 ′ = 4𝜋0 𝐼 ∫ 𝓇 2 {for surface and volume currents sub I for K(r’) and dl’ for J(r’) in above eq} Ampere’s Law ∮ 𝐵 ∙ 𝑑𝑙 ′ = 𝜇0 𝐼𝑒𝑛𝑐 ∫(∇ x𝐵) ∙ 𝑑𝑎 = ∮ 𝐵 ∙ 𝑑𝑙 = 𝜇0 ∫ 𝐽 ∙ 𝑑𝑎 ∇ x 𝐵 = 𝜇0 𝐽 ∇∙𝐵 =0 Vector Potential 𝐵 = ∇x𝐴 ∇∙𝐴=0 𝜇 𝐴(𝑟) = 4𝜋0 ∫ 𝐽(𝑟 ′ ) 𝓇 𝜇0 𝑚 x 𝓇̂ 𝐴𝑑𝑖𝑝 = 4𝜋 𝑑𝜏′ 𝑟2 Magnetic Dipole Moment 𝑚 = 𝐼 ∫ 𝑑𝑎 = 𝐼𝑎 𝜇 𝑚 𝐵𝑑𝑖𝑝 = 0 3 (2𝑐𝑜𝑠𝜃𝑟̂ + 𝑠𝑖𝑛𝜃𝜃̂) 4𝜋𝑟 CHAPTER 6 Torque 𝑁 = 𝑚x𝐵 Bound Currents 𝐽𝑏 = ∇ x 𝑀 𝐾𝑏 = 𝑀 x 𝑛̂ Ampere’s Law in magnetized materials 1 𝐻 ≡𝜇 𝐵−𝑀 0 ∇ x 𝐻 = 𝐽𝑓 ∮ 𝐻 ∙ 𝑑𝑙 = 𝐼𝑓𝑒𝑛𝑐 𝐵 = 𝜇0 𝐻 Permeability and Auxiliary Field 𝑀 = 𝜒𝑚 𝐻 𝐵 = 𝜇0 (𝐻 + 𝑀) = 𝜇0 (1 + 𝜒𝑚 )𝐻