Prof. David R. Jackson ECE Dept.

ECE 2317 Applied Electricity and Magnetism Spring 2014 Prof. David R. Jackson ECE Dept. Notes 28

Biot-Savart Law r J R r V r R r Js S Please see the textbooks for a derivation. a) Volume current b) Surface current

Biot-Savart Law (cont.) B r r R I C A c) Line current (wire) Note: Rule: The contour C is in direction of the I arrow (i.e., the reference direction for the current). (This determines the starting point A and the ending point B.)

Example z Find H r = (0, 0, z) R I y   r x z = 0 Infinite line current

Example (cont.) z r´ R I y  x r

Example (cont.) Note: Hence we have or Note: This agrees with the answer obtained by Ampere’s law.

Example z r = (0,0,z) R a y r´ x I Find H so Note: This problem cannot be solved using Ampere’s law!

Example (cont.) z r = (0,0,z) R a y r´ x I

Example (cont.) n= # turns/meter a z I Solenoid (revisited) Use result from single loop (previous example): a z “slice” at z

Example (cont.) Hence or a z

Magnetic Materials S S S S N N N N B Because of electron spin, atoms tend to acts as little current loops, and hence as electromagnetics, or bar magnets. When a magnetic field is applied, the little atomic magnets tend to line up. This effect is what causes the material to have a relative permeability. Please see the textbooks to learn more about magnetic materials and permeability.

Boundary Conditions Region 1 Ht1 Bn1 Js Bn2 Ht2 Region 2 The unit normal vector points towards region 1. Note: If there is no surface current:

Boundary Conditions (cont.) Ht Js PEC Assume zero magnetic field inside the PEC (This is true for any f > 0, please see the note at the end on skin depth). Note: The surface current is directly related to the tangential magnetic field at the surface.

Boundary Conditions (cont.) B PEC Magnetic field lines must bend around a perfect electric conductor (PEC). This is the opposite behavior of electric field lines. E PEC

Boundary Conditions (cont.) Note: The “skin depth”  of a metal determines how the magnetic field will vary within the conductor (discussed in ECE 3317). The field penetrates with distance z into the conductor as • Finite conductivity and zero frequency:  =  (uniform field inside conductor) • Infinite conductivity and nonzero frequency:  =0 (zero fields inside conductor)  , Conductor z

Magnetic Stored Energy We also have Hence we can write

Example Ls a z N turns I Find UH inside solenoid Assume infinite solenoid approximation.

Inductance S I The unit normal is chosen from a “right hand rule for the inductor”: The fingers are in the direction of the current reference direction, and the thumb is in the direction of the unit normal. Single turn coil The current I produces a flux though the loop. Note: Please see the Appendix for a summary of right-hand rules. Definition of inductance:

N-Turn Solenoid N I We assume here that the same flux cuts though each of the N turns. The definition of inductance is now

Example Ls a z I N turns FindL Note: We neglect “fringing” here and assume the same magnetic field as if the solenoid were infinite. so

Example Ls a z I N turns Final result: Note: L is increased by using a high-permeability core!

Example N turns I Toroidal Inductor Note: This is a practical structure, in which we do not have to neglect fringing and assume that the length is infinite. • Find Hinside toroid • Find L

Example (cont.)

Example (cont.) N turns I The radius R is the average radius (measured to the center of the toroid). Assume Ais the cross-sectional area: a A

Example (cont.) so Hence I We then have C r N turns

Example (cont.) N turns I a S R Hence

Example Coaxial cable Ignore flux inside wire and shield I FindLl (inductance per unit length) a I h b z A short is added at the end to form a closed loop. I I Center conductor - S + z h Note: S is the surface shaded in green.

Example (cont.) I I z Center conductor - S + h

Example (cont.) I a I h b z Hence Per-unit-length:

Example (cont.) Recall: From previous notes: Hence: (intrinsic impedance of free space)

Voltage Law for Inductor Apply Faraday’s law C C is chosen as counterclockwise. Hence  =flux going though the loop in the upward direction (out of the page). This follows from the RH rule in Faraday’s law for the unit normal: B + - v(t) Faraday’s Law (integral form) Faraday’s Law (differential form) Stokes’s theorem Note: C is a stationary path.

Voltage Law for Inductor C B PEC wire: + - v(t) Note: There is no electric field inside the wire. Hence

Voltage Law for Inductor (cont.) C B i + - v so Note: The direction of the current is chosen to be consistent with the RH rule for the inductor. or Note: We are using the “active” sign convention.

Voltage Law for Inductor (cont.) B i - + v To agree with the usual circuit law (passive sign convention), we change the reference direction of the voltage drop. (This introduces a minus sign.) Passive sign convention (for loop) Note: This assumes a passive sign convention.

Energy Stored in Inductor L + v - i t = 0 - + + v(t) - Hence we have

Energy Formula for Inductor We can write the inductance in terms of stored energy as: Next, we use We then have This gives us an alternative way to calculate inductance.

Example N a z Ls Find L Solenoid From a previous example, we have Energy formula: Hence, we have

Example FindLl I a I h z b Coaxial cable We ignore any magnetic stored energy inside the conductors.

Example (cont.) I a I h z b Per-unit-length: Note: We could include the inner wire region if we had a solid wire. We could even include the energy stored inside the shield. These contributions would be called the “internal inductance” of the coax.

Appendix In this appendix we summarize the various right-hand rules that we have seen so far in electromagnetics.

Summary of Right-Hand Rules Fingers are in the direction of the path C, the thumb gives the direction of the unit normal. Stokes’ law: The following two rules follow from Stokes’s theorem. Fingers are in the direction of the path C, the thumb gives the direction of the unit normal (the reference direction for the flux). Faraday’s law: (fixed path) Fingers are in the direction of the path C, the thumb gives the direction of the unit normal (the reference direction for the current enclosed). Ampere’s law:

Summary of Right-Hand Rules (cont.) Magnetic field law: For a wire or a current sheet or a solenoid, the thumb is in the direction of the current and the fingers give the direction of the magnetic field. (For a current sheet or solenoid the fingers are simply giving the overall sense of the direction.) Inductor definition: Fingers are in the direction of the current Iand the thumb gives the direction of the unit normal (the reference direction for the flux).

Prof. David R. Jackson ECE Dept.