Electromagnetism (2018 Edition)

Contents

Electromagnetism

		Law (about charge and electric field), the Ampere's Law (about current and magnetic field), and the Faraday's Law (a relationship between E and B), see Figure 02. Faraday's observation has inspired J. C. Maxwell to assemble these laws into a consistent set of equations in 1865 and is now known as Maxwell's equations. The disturbance of the electromagnetic fields was subsequently identified as the light wave in optics (see wave equations below).
Figure 01 Electro-magnetism	Figure 02 Maxwell's Equations	Electromagnetic Wave Equations :

EM (ElectroMagnetism) Basic

The origin of EM basic can be traced back to Newton's law of Universal Gravitation in 1686. It states that the force : FG = - GmM/r2, where m and M are the two point masses, r the distance between them, G is a proportional constant known as gravitational constant, which was determined by the Cavendish Experiment (Figure 03) in 1797 to have a value of 6.74x10-8 cm3/gm-sec2 (vs 6.67x10-8 cm3/gm-sec2 in 2014). Similarly, the force between two electric charges was postulated to have the form : Fe = keqQ/r2,

Figure 03 Gravitational Constant [view large image]

where q and Q are the two point charges, r the distance between them. The proportional constant ke was measured by torsion balance similar to the Cavendish Experiment.

Some comments about F_e :


The measured value of ke = 9x1018 gm-cm3/sec2-C2, where C = coulomb is the mks unit of charge. One coulomb is the collective charges of 6.25x1018 electrons. If we define 1 coulomb = 3x109 statcoulomb, then ke = 1, and the force simplified to F = qQ/r2 (with charges in unit of statcoulomb. Mathematical formulas throughout this text is referred to statcoulomb for simplicity). Inside intervening dielectric materials (electrical insulators that can be polarized by an applied electric field), the force between two charges becomes weaker but the capacity to store charge (capacitance) increases. In order to stress the strengthened charge capacity, it is often re-define ke = 1/4, where the 4 is inserted to get rid of the troublesome factor from the surface integral (Figure 02). In vacuum, = 0 = 7.96x10-3 sec2-2/gm-cm3. The is called permittivity, and K = /0 1 is the dielectric constant (Figure 04). The electric susceptibility e = K - 1 is for measuring the degree of polarization of a dielectric material in response to an applied electric field. For 2 particles with same charge q and mass m, the force ratio of electric to gravity is Fe/FG = 1.47x107(q/m)2(gm/)2. For electron with e/m = 5.28x1017 /gm, Fe/FG = 4.18x1042 - evidently the gravitational force is negligible compared to the electro-static force. On the other hand, the Earth and Moon are electrically neutral with q = Q = 0; in that case gravity becomes overwhelming.

Figure 04 Some Dielectric Constant K

Unlike gravitational force, which is always attractive leading to the formation of black holes, the electric force is attractive only between positive and negative charges. The repulsive force between like charges is

responsible to maintain an equilibrium configuration for all kinds of objects (see potential curves for H₂ molecule). However, for the case of stopping the gravitational collapse of a star such as the white dwarf, the force or pressure responsible is from a quantum mechanical process by the title of "Pauli Exclusion Principle" (see Degeneracy Pressure, and a "Diagram of Cold Dead Matter" illustrating the various processes to resist the crushing gravity).

	The same kind of mathematical scheme has also been applied to the case of force per unit length between two infinitely long, parallel wires separated by a distance r and carrying currents I and I' : dFm/dl = 2km(I I')/r (see Figure 05), where I and I' are in unit of statampere (amp) = 1 /sec (one unit of charge passing a given point per second).
Figure 05 Currents Interaction [view large image]	By comparing the magnitude of Fe and dFm/dl for known charges and currents, it is found that ke/km = c2 (velocity of light c = 3x1010 cm/sec).

		The electric field E of a point charge Q can be defined by re-writing the force Fe = qE, in which, E = (Q/r2) is the electric field by the charge Q in unit of /cm2. The electric potential is defined by : . A test particle with charge q would have a potential energy EP = qV (see Figure 07, both V and Ep are scalars while E and Fe are vectors). V = EP/q is the voltage, i.e., potential energy per unit charge. It is always measured across 2 points in the form of difference, i.e., V = Vab = Va - Vb = I R :
Figure 06 Permeability List [view large image]	Figure 07 Field and Potential

Inside dielectric medium, the electric field is replaced by the displacement field D = KE to take care the unknown composition within the material (the K here is the dimensionless dielectric constant, but denoted by most textbooks as - a rather confusing practice). Similarly for the magnetic field B inside diamagnetic (r < 1, repulsive) and paramagnetic (r > 1, attractive) materials, it is replace by the magnetizing field H = rB (most textbooks would use instead of r). Ferromagnetic material retains magnetization (induced magnetic dipole) after the external field is removed (Fogire 08).

Figure 08 Ferromagnetization [view large image]

Electromagnetic wave is the propagation of the vibrational E and B fields : .

Coulomb's Law

			charge density , i.e. E = (Figure 09b and see Figure 10 for some formulas of vector analysis in three different coordinate systems). Sometimes the solution can be obtained readily by integrating the divergence over a volume. As shown in Figure 11, the Gauss' Theorem transforms the volume integral to a surface integral, from which some examples for its
Figure 09 Coulomb's Law [view large image]	Figure 10 Vector Formulas [view large image]	Figure 11 Gauss' Theoerm	application are shown in the followings. The surface enclosing the volume is called Gaussian Surface.

Point Charge, Prefect Conductor, and Capacitor -

Figure 12 Gaussian Surface [view large image]	The factor of 40 disappears in the spherical surface integration because we adopt the statcoulomb as the unit of charge (see comments on Fe).	Figure 13 Capacitance [view large image]

Multipole Expansion -
The general solution for the electric field E as shown in Figure 09 seems to be intractable for arbitrary distribution of charges. The method of multipole expansion for the potential defined by E = - enables extraction of some information depending on the level of detail. Since

Figure 13a Multipole Expansion [view large image]

Ampere's Law

		The curl of a vector function A, i.e., (A) is about going around a looped path as shown in Figure 14, which illustrates the process with the four sides of a rectangle on the y-z plane. A little bit more imagination would be able to generalize the concept to 3-D space (see Figure 10 for the expressions in three different coordinate systems). The Ampere's Law identifies the vector function to be the magnetic field and the change is related to the source/sink of electric current passing through the surface enclosed by the loop, i.e., B = J/c, where J is the current density in unit of C/cm2-sec. This is in analogy to the "divergence" in which the vector function goes through a closed surface and the source/sink is the charge within the enclosed volume. So the Ampere's Law is prescribed in a formulation one dimension lower and its corresponding integral form is performed on a surface enclosed by the loop (see Figure 02) and derivation below.
Figure 14 Curl of Vector [view large image]	Figure 15 Stokes' Theorem [view large image]

Here's a few examples :
• Magnetic Field by a Straight Line of Current - The solution is obtained by using this integral form of the Ampere;s Law around a circular path of radius r as shown in Figure 16,a. The constant B along the path is B = I/2rc. The force per unit length between two parallel currents I and I' is dF/dl = (I'B)/c = (I' I)/2rc² (see Figure 05).

Magnetic Field by Circular Current Loop - The B field at the center of the loop can be calculated via the differential form B = J/c. In cylindrical coordinates (see Figure 10), the only surviving term in the direction of the current density is dBz/dr = J/c. Integrating from r = 0 to R yields Bz = JR/c. Since J = I/R2, B = I/Rc. The expression on the z axis is more complicated as shown in Figure 16,b. For z >> R, such B field is similar to bar magnet or magnetic dipole, i.e., B = M/z3, where the dipole moment M = IR2/c .

Figure 16 Ampere's Law, Examples [view large image]

The off axis formula is even more involved (see "Simple Analytic Expressions for the Magnetic Field of a Circular Current Loop").

Such configuration is useful for many applications in which a current is used to establish a magnetic field as in an electromagnet or a transformer. These kinds of devices usually would have n turns of loops closely spaced with B = nI/Rc at the central axis .
• Displacement Current - This is the I_c(t) defined in Figure 16,c. Instead of moving electric charges, it is generated by time varying electric field. The idea was conceived by James Clerk Maxwell in 1861, initially also involved the polarized charges of the dielectric material. This term is indispensable in deriving the electromagnetic wave equations, which is obtained by operating
  () and (/t) on the Ampere's Law, and Faraday's Law (Figure 02), then running some manipulations with the help of the identity A = -²A + (A). Finally, we obtain :      
  
  See inserted diagram, the Ampere's Law, the formula for A in Figure 10 and note the reversed direction of the z-axis (as opposed to the normal convention), also the wonder of "Electromagnetic Spectrum".

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  Gauss's Law (Absence of Magnetic Monopole)

  It should be labeled as Gauss's Law for magnetism. Simply put, the divergence of the magnetic field B is always zero, i.e., B = 0. There is no such thing as magnetic charge although there are plenty of magnetic dipoles. Conceptually, both the E / B fields and qe(electric charge) / qe(magnetic charge) are respectively components of a vector in a 2 dimensional EM space (Figure 17). The rotational transformation between the components leaves the Maxwell's equation unchanged (i.e., invariant, see "Magnetic Monopole"). In the real world, such symmetry is somehow broken. This is very strange for someone who believes "Physics is Beautiful". Theoretical physicists

  Figure 17 EM Space [view large image]

  have pondered and experimentalists have searched for its existence without success for 2 centuries. Somehow this law has to be there, or the electromagnetic wave equation would not have its current form (see above).

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  Faraday's Law

  According to the definitions of magnetic flux and elecromotive force (EMF) in Figure 18,a, the Faraday's Law can be recast as
  - = (1/c)d/dt. However, "elecromotive force" is a misnomer as it is not a vector like the force; it is actually a scalar similar to energy.

  	The backward EMF actually acts like a time varying voltage, i.e., - = VL, oscillating between positive and negative according to whether the flux is increasing or decreasing, i.e., d/dt > 0 or <0. There are many ways to change , the most common method is to rotate either the coil or the magnet (Figure 18,b and c). All processes involve the transfer of mechanical to electrical energy.
  Figure 18 Electromotive Force [view large image]	Here's a few examples to illustrate the process and its applications.

		Electric Inductor - It is one of the three basic elements in electrical circuits. The inductance of such device is defined as L = /I. It can be considered as the proportional constant for = L I thus d/dt = L dI/dt. The L also depends on its construction such as the relative permeability of the core material r, the number of turns N, the area A and lehgth , i.e., L = rN2A/c in cgs unit of sec.
Figure 19 Inductor in DC [view large image]	Figure 20 Inductor in AC [view large image]	Then = -VL = -(1/c)d/dt = -(L/c) dI/dt, where VL is the voltage across the indictor. The variation of VL in a DC circuit with load R is plotted in Figure 19.

It shows that V_L changes in a diminishing attempt to oppose the rising current and finally subsided when there is no more change in current or flux to support it. Figure 20 shows the behavior of an inductor in AC circuit without any load but running with a alternate voltage generator.



LC Circuit - This circuit has only the elements of inductor L and capacitor C with no voltage source, the sum of the voltages across them would be VL + VC = 0. It's a differential equation in terms of the charge Q : (L/c)d2Q/dt2 + Q/C = 0. This is in the form of standard harmonic oscillator, the solution of which is : Q = Q0 cos(0t). Differentiating it once yields I = I0 sin(0t), where I0 = 0Q0, and 0 = (c/LC)1/2 is the resonant frequency.

Figure 21 LC Circuit [view animation]

If we inject either charge to C or magnetic flux to L, current will oscillate back and fore in the circuit forever as shown in the animation of Figure 21. This feature is used most commonly in tuning radio transmitters and receivers. There is no prefect conductor with no heat loss, the equation in the real world is the next example.

• RLC Circuit - It is similar to the LC circuit just by adding a resistor R and a voltage source for maintaining the flow of the oscillating current. In term of the charge Q, the equation takes the form :
  (L/c)(d²Q/dt²) + R(dQ/dt) + Q/C = V(t).
  In term of current I and recasting into a form familiar to harmonic oscillator, it becomes :
  (d²I/dt²) + 2k_d(dI/dt) + k_sQ = (c/L)dV(t)/dt, where
  k_d = (cR/2L), (k_s)² = (₀)² = (c/LC).
  The general solution is :

  	For AC voltage source V = V0sin(t) with arbitrary frequency , the solution takes the form : I = V0 sin(t-) / {R2 + [L/c-(1/C)]2}1/2 , where = tan-1{[(L/c)-(1/C)]/R}.
  Figure 22 RLC Circuit [view large image]	Z = {R2 + [L/c-(1/C)]2}1/2 is called the electric impedance. It is the extension of electric resistance to AC circuit. The stringing sequence of these electrical elements is called series connection.

  
  
  Some limiting cases :
  
  
  1. In the series connection, the current I is the same in all parts of the circuit, while the voltage across each element would be different, e.g., V = V_R + V_L + V_C .
  
  
  2. For V = 0, dV/dt = 0, R = 0, k_d = 0; b = i₀.
     Selecting the sine function from e^i₀t = cos(₀t) i sin(₀t), I = Asin(₀t) + B.
     The initial condition for t = 0, I = 0 B = 0, while t = /2₀, I = I₀ A = I₀.
     Finally, we obtain I = I₀sin(₀t). (see resonant circuit in Figure 21)
  
  
  3. For V = V_b = constant, dV/dt = 0, C = for no capacitor, k_s = 0; b = -2k_d = -cR/L.
     I = Ae^bt + B.
     The initial condition for t = 0, I = 0 A = -B, while t = , I = V_b/R B = V_b/R.
     Finally, we obtain I = (V_b/R)(1 - e^-t/), where = L/cR. (see inductor in DC circuit Figure 19)
  
  
  4. For V = V₀sin(t) with arbitrary , C = for no capacitor, R = 0, k_d = 0.
     I = (V₀c/L)sin(t-/2) = (-V₀c/L)cos(t) (see inductor in AC circuit Figure 20)
  
  
• Electric Generator - The principle of operation for most of the electric power generators is based on the electromotive force (EMF) in the Faraday's Law. The two types mainly used in modern time are just the glorified versions as illustrated in Figure 18,b and c. Followings are some of the features and terms.

  Figure 23 Electric Generator, DC [view large image]	Figure 24 Electric Generator, AC [view animation]	Figure 25 Electric Generator, Types [view large image]	Figure 26 Electric Generator, Hydro [view large image]

  • Rotor - The rotating part of the electric generator. It could be the magnet or the coils.
  
  
  • Stator - The stationary part of an electric generator, which surrounds the rotor. It could be the magnet(s) or the coils.
  
  
  • Armature - The power-producing component, i.e., the coils.
  
  
  • Field Magnet - It could be a permanent magnets as in magneto or electro-magnet in dynamo (DC generator) and alternator, which dominates large scale AC power generation in modern day.
  
  
  • Commutator - This is designed to produce DC current by reversing the contact point every half turn. As shown in Figure 23, the EMF varies as the coil rotates. It is the commutator which reverses the negative to maintain an uni-directional but time varying voltage.
  
  
  • Three-Phase AC Generator - There are three pairs of stationary coils arranged 60^o apart producing three output in three different phases all in same frequency and magnitude by a rotating magnet (see Figure 24). It is the most common AC genrator used by electrical grids worldwide. Common households only use one pair. However all three pairs are connected to an electric stove or dryer (see here ) for more power supply.
  
  
  • Balanced Load - As shown in Figure 24, the currents or EMFs of the 3 phases always added up to be zero. Therefore the neutral line

    should carry no current. Unbalanced currents (between the 3 phases) would cause erratic operation, over heat, and low efficient. In most cases, the electrical company (supplier of the three-phase power voltage) would fed each customer from one of these phases and the neutral, different customers in the same area are connected to different phases, but share the same neutral. The number of connected properties are arranged such that each phase has a load similar to the rest, that is, the load is balanced (or almost balanced) for the three-phase system with no returning current in the neutral wire (Figure 27a).

    Figure 27a Three Phase Power Distribution

    
    
  • Parallel Connection - The single phase AC input to the house is at 60 Hz, 120 volt. The loads are arranged in parallel so that the voltage received by each load is the same while the currents flow through them may be different (see Figure 27b for single phase AC). Thus,

    	I1 = Vab/R1, I2 = Vab/R2, I3 = Vab/R3. As there is no accumulating of charge in the circuit, it follows that I = I1 + I2 +I3 at the exit point b (same as at the entry point a). Thus, I = Vab/R, where 1/R = 1/R1 + 1/R2 + 1/R3 is the reciprocal of the equivalent resistance. If one of the load has short circuit, e.g., R2 = 0, then I tripping the circuit breaker or blow a fuse.
    Figure 27b Parallel Connection	In US and Canada, the live wire is in red or black, the neutral is white. The ground wire (in green color) is connected to the frame of the load, it should carry no current unless there is leakage of electricity.

    
    

    Turbine - This is the part that supplies mechanical energy to the system. It is fit with blades to absorb the impact from the incoming fluid, and wicket gates to control the amount and thus the input energy (Figure 27c). The fluid can be steam for the coal/nuclear plant both of them are known to produce harmful residuals (see for examples, "Coal and Air Pollution", and "Nuclear Waste"), or water in hydro-electric plant, which may be cleaner but ruins the environment by blocking river flow (Figure 25, also see "Hydropower and the Environment" ). Figure 26 show some details of a hydropower plant and the other components of the electrical grid such as the transformer and transmission line.

    Figure 27c Turbine [view large imgae]

• Transformer - This is another application of the Faraday's Law of induction by winding two separated coil with winding number N_p (for primary) and N_s (for secondary) through a common steel core (Figure 28). Since there is a common passage for the magnetic flux , we have
  V_p=(N_p/c)d/dt, V_s=(N_s/c)d/dt; or V_p/N_p = V_s/N_s.
  Conservation of electricity power on the electric circuit gives :
  I_pV_p = I_sV_s, which yields a chain of formulas : V_p/V_s = I_s/I_p = N_p/N_s = (L_p/L_s)^1/2 = a and finally, V_s = (N_s/N_p)V_p = V_p / a .
  For step-up the voltage in the secondary a = (N_p/N_s) < 1; while a > 1 is for step-down.

  		This is the basic for the principle of transformer operation which becomes indispensable for many applications. The formulas are for ideal condition with no heat loss. This is not the case in the real world as witness by the warm touch of the thing. Figure 29 lists a few of its daily usages. The very important application is on the electricity transmission.
  Figure 28 Transformer [view large imgae]	Figure 29 Transformer Usages [view large imgae]	But its most ubiquitous use is in the AC/DC adapter, in which the transformer steps down the voltage and then restricts the current to flow in uni-direction by a rectifier made with diode.

• Power Transmission - The source of power supply is usually faraway from its consumers. It can be shown that the power available at the point of consumption is : P_R = [R/(R+R_C)]P_V, where P_R is the useful power with resistance R, P_V is the power supply, and R_C is the total resistance of

  	the transmission line, which can be very significant. Now if the voltage is step-up at the supplying side and then step-down at the point of consumption with the ratio a = (Np/Ns) as stated above; then the power formula becomes PR = [a2R/(a2R+RC)]PV PV, if a2R >> RC, that is, almost no power loss during the transmission (see Figure 30 for such arrangement).
  Figure 30 Power Transmission [view large imgae]

  		Electric Motor - This is the reversed conversion of electrical energy into mechanical energy (Figure 31). Instead of mechanically varying the magnetic flux to induce current, the electric motor uses current flow in a magnetic field to generate motion according to the Lorentz force as illustrated in Figure 32. Its applications includes industrial fans, blowers, pumps, machine tools, household appliances, power tools, disk drives, ...
  Figure 31 Electric Motor, DC [view large imgae]	Figure 32 Lorentz Force [view large imgae]	and now comes the electric car. It has been promoted as the green technology good for the environment. It only requires plug-in to the AC

  outlet for replenishing the battery and no more tail-pipe emissions. But ultimately, it all depends on how the AC power supply is generated. It could be from a coal or nuclear power plant causing more problems at another level. Most "Electric Car Pro and Con" articles do not take such consideration into account.
  
  
• Electrical Safety - In an emergence of electric leakage, it is advisable to avoid touching object of low electrical resitivity (in MKS unit of ohms-m or -m), which has a specific value for different material (see Table 01). It is related to the electrical resistance R = /A, where A is the cross-section, and the length (it is more practical to use MKS unit on subject of engineering).
  

  Table 01 List of Resitivity

  It seems that human skin has very high electrical resitivity. However, it is rather deceptive; the protection could be lower by factor of ~ 100 when it is wet, the skin can be burnt off or cut open, and dielectric breakdown would occur when the voltage is above 450 - 600 V (see other causes in "Electric Injury").
  
  

  The effect of electric current in human body depends on the environment (wet or dry), the posture of contact ( path of current flow inside the body), and the voltage and electric resistance R = /A of the body. Figure 33a shows the effect of current in terms of its magnitude, Figure 33b shows the resistance values of different body parts. For example (in Canada), the household AC voltage is 120 V, for the averaged body resistance of 1000 ohms, the current flow through the body would be I = 120/1000 = 0.12 Amp = 120 mA, which could be fatal if the flow lasts more than 1 second according to Figure 33c. The damaging voltage is about 30 volts for 30 mA to cause suffocation (Figure 33,a).

  Figure 33 Resistance, Human Body [view large imgae]

  
  
  Here's some comments on protection against faulty electrical equipment (also see illustrations in Figure 34) :

  (a) Electric current causes damage via collisions of energetic electrons with molecules inside the body. It is determined by I = Vab/R, where I is the current, the voltage Vab is fixed by the power grid usually at 120 or 240 AC volts; R the electric resistance depends on the intrinsic property of the material and its geometric shape (see Table 01). Thus the higher value of the resistance on a specific path of the circuit would reduce the current to lower value - minimizing the damage.

  Figure 34 Protection against Faulty Electric [view large imgae]

  (b) As stated previously in "Parallel Connection", all the loads inside the household are connected in parallel with a fixed voltage Vab supply of 120 volts (in Canada).

  The current flow through each load is I₁ = V_ab/R₁, I₂ = V_ab/R₂, I₃ = V_ab/R₃, ... with the total current I = I₁ + I₂ + I₃ + ... The equivalent resistance for the entire circuit is R = 1 / [(1/R₁)+(1/R₂)+(1/R₃)+...], which shows that the lower resistances are the dominant contribution. In case of short circuit in one of the loads, say R₂ = 0, then R = R₂ = 0, I = V_ab/R = .
  
  
  • (c) If one of the loads (for example a kettle) sustains a faulty connection to its case, then the case becomes part of the live wire. The kettle may still be working with a load of about 60 and a current I_kettle = 120/60 = 2 Amp.
  
  
  • (d) A body (with averaged R_body ~ 1000 ) in touch of this faulty kettle case would receive a current of I_body = 120/1000 = 0.12 Amp = 120 mA, which is enough to stop the heart from beating (see Figure 33,a). The returning path of the current in this case is via the Earth, which is a moderate conductor (see Table 01).
  
  
  • (e) If this circuit is installed with a circuit breaker of 15 Amp, then it requires a grounded wire with R_ground = 120/15 = 8 or less to trip off the breaker. The grounded wire should be at least 2.5 meters deep and preferably below the frost line having a resistance of 5 or less. It provides a common reference point, e.g., V_b = 0, between the current source and sink.
  
  
  • (f) There are even more sensitive safety device called "Ground Fault Circuit Interrupter (GFCI)", which stops the current flow if the current in the live wire is different from the neutral. Using the kettle as an example, in normal operation I = I_kettle = I_neutral; but in case of a faulty connection I = I_kettle + I_ground + I_body = I_neutral + I_ground + I_body I_neutral .
  
  
  • (g) Wearing insulated shoes of very high resistance (rubber shoes for example, see Table 01) would also prevent electrical injury; since I_shoe = 120/R_shoe 0 as R_shoe .
  
  
  • (h) The huge current surge from lightning strike is diverted to the ground. This is the concern of voltage transformers, which are mostly located outdoor.
  
  
  • (i) The most dangerous circumstance is around the downed (very high voltage) power line. Stay back at least 10 meters even if it is not sparking. See "Home Safety - General Information ".
  
  
  • (j) Safest measure is to stay away from electrical hazard, real or suspected especially for young children. See "Electrical Safety for Kids".