Stationary electric field in conducting medium The stationary
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Stationary electric field in conducting medium
The stationary electric field in conducting medium will be considered. The constant current is flowing in this conducting material. This field is described by two vectors E - J - current density γ- i electric field intensity l the conductivity of the conductor ΔSn
Equations of electric field in conducting medium Differential form integral form
Generalized Kirchhoff’s laws KCL – the current flowing through a closed surface =0 S i 1 i 5 i 4 i 2 i 3 J 4 d. S 4
KVL – the sum of the voltages along any closed curve =0 C Et dl E
Ohm’s law formulated in field terms Vector form of Ohm’s law Let’s derive Ohm’s law, which expresses the proportionality of the current flowing through the resistance and the voltage across them. R
Joule’s law Electric energy transforms into heat in the conductor. φ ΔS ΔVol. Δl φ+Δφ ΔS p – the volume power density
Joule’s law in „field form” The continuity of vectors E and J components E 1 E 2 E 1 t=E 2 t J 1 n=J 2 n
Electric field OHM’s law KCL KVL JOULE`s law Analogy: Electric circuits
Stationary electric field in conducting medium Resistance of transition i -i B A MET
Problem: Let’s calculate the resistance of transition Rt of the cable isolation. The cable is placed in the ground. i R 1 R 2
Let’s assume the cable length l=1 m or l=1 km, then we receive the resistance per unit of cable length The capacity between the working wire and the screening coat or ground is the same as for the cylindrical capacitor. The parameter characterising a cable is to be found in cable catalogues:
Resistance of an isolated electrode: When we move the electrode B to infinity:
Earth electrodes I A R r B v. AB
A. Current density I R B. Electric field intensity C. Potential with respect of the reference point r
φ(r) φ(R) v. AB R r. A r. B r
We can calculate the radius of dangerous zone around the electrode from the inequality: Potential of earth electrode:
Earth electrode buried very deep. i
Spherical earth electrode dug in under the ground surface on a finite depth h. i 1 i h + h h h i i 2
Let’s assume that upper medium is nonconducting i h then There is only one model because in nonconducting medium this field doesn’t exist. h i
A deep pit has appeared near the electrode a a i i h h i h i i a a a
Magnetostatic field exists around permanent magnets and wires with the constant current. Equations of magnetostatic field: I
Fundamentals of Magnetostatic Fields • The effects of magnetic fields known for alomost three milllennia – certain stones attract iron – a large deposit of these stones (“lodestones”) found in the district of Magnesia in Asia Minor (Fe 3 O 4) – first scientific study written in 1600 by William Gilbert – early nineteenth century Hans Christian Oersted an electric current in a wire affected a magnetic compass needle – Oersted, Ampere, Gauss, Henry, Faraday, and others raised the magnetic field to equal partner status with the electric field (confirmed later by Maxwell)
Basic properties: (1) • Cutting of a large permanent magnet creates a number of smaller permanent magnets • A magnetic monopole has not yet been observed to exists in nature impossible to separate the north pole from the south pole of magnet
This field is described with the pair of vectors: B and H Definition of magnetic induction results from the Lorenz force:
Definition of B For defining the vector B we assume that only magnetic field exists, E=0. Because B is the vector it has a length and a direction, hence its definition has parts. The vector B numerical value is given by: Vector B direction is determined by such vector of charge velocity, where there is no force acting on the charge. B is the magnetic flux density (inductance), the SI unit: tesla (T).
Definition of magnetic field intensity (or magnetic field) is the result of magnetization phenomenon. • Moving electron circulating about positive nucleus can be modeled as small electric dipole • It can be also interpreted as being current hence (Oersted theory) the atom can be thought of as being small magnetic dipole • These dipoles will be assumed to be oriented randomly :
What will happen to these magnetic fields of individual atoms if external magnetic field is applied to the material? It depends on the type of considered material. Def. is the vector of magnetization magnetic flexibility or susceptibility
Three classes of materials: • Diamagnetic materials – Bismuth, copper, diamond, , gold, silver, silicon, lead, mercury • Paramagnetic materials – Aluminium, oxygen, magnesium, titanium, … • Ferromagnetic materials – iron, nickel, cobalt
Diamagnetic materials • The magnetic dipoles get reoriented such that their magnetic dipole moments m are in slight opposition to the applied magnetic field B • Without external field magnetic moments of rotating about the positive nucleus cancels magnetic field creating by the spin of the electron • External field perturbs the velocities of the orbiting electrons small magnetic moment for the atom is created (Lenz’s law) opposite to applied field.
Paramagnetic materials • Without external field magnetic moments of rotating about the positive nucleus do not cancel magnetic field creating by the spin of the electron completely leaving the atom with small net magnetic moment. • External field tends to align these moments in the direction of the applied field.
Ferromagnetic materials • Domains containing 1015 atoms, each has all of the magnetic domains oriented in the same direction (without external field) • Net magnetization = 0. • With external field domains having their magnetic fields aligned with the external one grow at the expense of the other. – If the external field is small this process will reverse itself – Strong enough field the domains rotate in the external field direction and collective direction of the domains will remain fixed.
Differential form Integral form
Ampère’s circuital law (or flow law or Ampère's law) This current is called a flow S c i 1 c 1 i 2 i 3 The turn of line c (right side of equation) and the signs of currents on the left side must be in agreement. When we turn the right-hand screw into the line c than we will write currents flowing according to the turn motion of the screw with „+”.
André-Marie Ampère 1775 - 1836 Author of the circuital laws
Magnetic field caused by very long straight wire with the current i Cylindrical system of coordinates will be used because this problem has a cylindrical symmetry. We will prove that vector i R r has only one component
Field doesn’t depend on the angular coordinate. It is obvious result of cylindrical symmetry. hence So Maxwell’s equation has the form:
Conclusion: The current flowing in a very long streight wire generates around this wire magnetic field which has only one component • perpendicular to the direction of the current density vector J, • tangential to the circles concentric with the wire. i R Only one component of the field Hθ depends on one variable r. exists. r
Problem: determine field H distribution for i R r H
H(r) Hmax r R
(*) Def. The notion: is called magnetic flux. Equation (*)expresses Kirchhoff’s law for magnetic flux: the sum of magnetic fluxes passing through the closed surface equals zero.
S
Electrodynamic force ΔS ΔV J Δi Δl
Electrodynamic force is acting on that element of the wire which is situated in magnetic field (determined by the induction B)
l dl B i Magnitude of this force is:
Let’s consider two very long streight parallel wires with the currents i 1 and i 2. The distance between wires is a. F 12 i 1 H 12 i 2 H 21 F 21
H 12 i 1 i 2 F 12 F 21 H 21
Let’s calculate the force acting on 1 m of two wires with the current 1 A situated in distance 1 m one from the other. Data: l=1 m, a=1 m, i 1=1 A, i 2=1 A
1 Amper definition: 1 A is the current which flowing in two infinitely long streight wires, with the small circle cross section, parallel, distanced of 1 m from each other, causes their mutual interaction on each metre of their length with the force
Data: a=20 cm z=400 b=50 cm Bgap=0. 3 T c=4 cm =1 mm Magnetic circuits Example of magnetic circuit: z – coils number B A C i z b d c E D a average way of the flux a F
Let’s calculate exciting current, which asserts the required induction value B=0. 3 T in air-gap The characteristic of magnetization for a ferromagnetic is given as a plot or as a table. 0. 1 0. 3 0. 4 0. 5 0. 6 1. 2 1. 35 1. 8 1. 6 0. 7 1. 1 1. 2 1. 4 1. 5 6. 75 9. 6 21. 9 30 B H
B A i z C b d c. F E D a a Circuit analogue: Φ 1 III I Φ 3 A cross section is equal S 2 in central column only, for the other intervals of the magnetic circuit the cross section is S 1. Φ 2 iz II Air-gap
1. Induction and field intensity in the gap: Air-gap magnetizes linearly. 2. The induction in right column is the same as in the air-gap, because the flux is the same.
Right column – index 3 Induction in the core is B 3=Bgap=0, 3 T The field intensity in this column will be read from characteristic: B H 3=1, 1 A/cm B 3 H H 3 Magnetic voltage across the central column: H in the air is large. Magnetic voltage is expressed in AMPERS
B A i z C b d c. F E D a a Circuit analogue: Φ 1 III I Φ 3 A cross section is equal S 2 in central column only, for the other intervals of the magnetic circuit the cross section is S 1. Φ 2 iz II Air-gap
3. Now we can calculate the field intensity H 2 in central column. B 2 will be read from characteristic of magnetization: B B 2=1, 2 T B 2 H H 2 So the flux in the central column:
4. The flux in the left column is the sum: And next induction B 1 : H 1 can be read from the characteristic of magnetization B H 1=21, 9 A/cm B 1 H H 1
5. The magnetic voltage across the left part of magnetic circuit: The flow in exciting coil : The current in exciting coil:
POTENTIALS OF MAGNETIC FIELD a. b. Vector potential Scalar potential
Magnetic field can’t be expressed as a gradient of scalar potential (like it was in case of electrostatic field) because Field is rotational (it is different from the electrostatic field where rot E=0).
SCALAR MAGNETIC POTENTIAL Let’s assume that in considered area current doesn’t flow. Analogically like in electrostatic field we can introduce scalar magnetic potential defined as:
Scalar potential satisfies Laplace’s equation
Let’s calculate vector potential: i R r H around the wire with the current using scalar
i R r H We have to determine constants C 1 and C 2. V=const
1. Let’s assume a reference point (a reference half-plane) position: 2. From Ampère’s law:
1. Let’s assume, that a reference point (the set of reference points) is in the half-plane θ=0. 2. From Ampère’s law:
Vector magnetic potential
Definition of vector potential is a consequence of equation and vector identity Definition of vector potential: Def. The condition of introduction of vector potential definition is div Hence: =0
This definition is ambiguous because and from vector identity: We say, that vector potential is determined with accuracy to the gradient (for a potential field) The additional condition is necessary for the vector potential to be unique. This is a condition for div. A.
Differential equation for a vector potential Maxwell’s equation: Assumption: μ=const. Let’s use the vector identity: ?
=0
J d. V P(x, y, x) r 0 V P 1(x 1, y 1, z 1) V – the volume with current P – observation point P 1 – source-point
In internal points potential satisfies vector Poisson’s equation: In external points potential satisfies vector Laplace’s equation: Vector equation is equivalent to three scalar equations: or
In general case the solution has a form: where r 0 – a distance between P and P 1 The integral is calculated with respect of source-point coordinates P 1(x 1, y 1, z 1)
The solution for the thin wire with the current i: d. S i d. V dl
when μ=const. Let’s calculate the magnetic field generated by the current in thin wire.
Magnetic field is calculated in point observation P(x, y, z) Rotation is calculated with respect of coordinates of point P. . The integration is made over the area V with the current so the point P 1 coordinates are variables of integration (x 1, y 1, z 1). Conclusion: Operations are independent.
=0 where
r
After all derivations we receive: The Biot-Savart law dl P 1 i c r P d. H
How to use this formula? Is it truth: The magnitude can be calculated from the scalar term: and vector direction can be determined from the right-hand screw rule? ?
When can vector sum be replaced with the algebraic sum? When vectors d. H are directed along one straight line. When does it happen? dl P 1 i c It happens when the wire and point P are laid in one plane. r P d. H
Let’s calculate the magnetic field in point P. P dl R i
Let’s calculate magnetic field due to the current i in the finite length wire. The distance from point P to the wire is a. i r l a P dl
Problem: Well known formula for very long, streight and thin wire with the current i will be derivated from Biot-Savart law. i r P 2
Electromagnetic induction
The magnetic field generates forces acting on charges which can move in the conductor. The effects of this forces action: • • The voltage appears across the open circuit (electromotive force EMF) The current is flowing when the circuit is closed (induced current) There are two ways of generating electromotive force : • the charge moves in magnetic field • the induction B passing through the circuit changes
Maxwell’s II equation This is Faraday’s law of electromagnetic induction What does this equation in integral form tell us?
There is an element of equation called EMF. Its value depends on the changes of magnetic field velocity.
The voltage induces between the coil terminals when the flux passing through the winding changes. Let’s consider a closed circuit placed in magnetic field changing in time. c e B
The Lorentz force acts on the charge. FB +q B v Let’s consider forces acting on a single charge q inside of the wire which is moving in magnetic field.
This force causes charges relocate. New charges distribution will generate new electric field Its action on the charge will balance the force FB
e. B Def: EMF of electromagnetic induction in thin wire :
Circuit scheme of this phenomenon: e. B V A value of e. B depends on the direction of integration, that is on assumed voltage arrow. EMF can be determined with RIGHT-HAND rule.
A A u. AB dl v v B dl B B B u. BA
Problem: A metal bar with the length l rotates with the angular velocity ω around the axis passing through one of its ends. Determine the voltage across the bar ends. B ω u r
B ω u r
Maxwell’s II equation This is Faraday’s law of electromagnetic induction What does this equation in integral form tell us?
There is an element of equation called EMF. Its value depends on the changes of magnetic field velocity.
The voltage induces between the coil terminals when the flux passing through the winding changes. Let’s consider a closed circuit placed in magnetic field changing in time. c e B
e - RHR F - LHR Lenz’s law v e e F F B i v B i
Lenz’s law 1. The current induced in the closed circuit moving in the magnetic field generates forces counteractive this moving. 2. The current induced in the closed circuit by the changes of magnetic field passing through the circuit generates own magnetic field counteractive the field changes.
Let’s consider a closed circuit placed in magnetic field changing in time. c e B
Self inductance • Let us consider electric circuit with current i 1 2 3 z
Denotes the magnetic flux passing through the surface bounded by the curve being the axis k-th turn of the circuit (coil) Algebraic sum of all the fluxes coming from all the turns of the circuit is called flux associated with the circuit 110
Definition • The coefficient of circuit self-inductance L (or simpler circuit inductance L) is the coefficient of proportionality from the formula: Associated flux Circuit current
Example: Wire inductance
a) Inductance from wire field
b) Inductance from cable insulation field
Total inductance of the cylindrical wire
Mutual inductance Coil 1 coil 2 116
Coil 1 Coil 2 Wd. WI 2015 PŁ 117
Definition Main flux Coil 1 Coil 2 leakage flux
Definition • The coefficient of mutual inductance M 21 of two coupled circuits with currents i 1 and i 2 is the coefficient of proportionality from the formula: Flux associated with second coil generated by the first one Circuit 1 current
Definition • The coefficient of mutual inductance M 12 of two coupled circuits with currents i 1 and i 2 is the coefficient of proportionality from the formula: Flux associated with the first coil generated by the second one Circuit 2 current
General formula
Maximum value of M Flux of coil 1 Flux of coil 2 generated by the coil 1 For maximum M fluxes are the same: Flux of coil 1 generated by coil 2 Flux of coil 2
Self inductance of the coil having z turns. 1 2 3 z Area of the cross section of the core Length of the coil core
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