ELECTROMAGNETISM

Introduction. The magnetic effect of current electricity was discovered by the Danish physicist Hans Christian Oersted (1777-1851) in 1819. He noticed during the course of a demonstration lecture that a compass needle, which happened to be near a wire connected to a battery, was deflected whenever the switch was closed so that the electric current flowed in the wire. He had been looking for such a deflection of the compass needle since he had noticed that sometimes a compass needle moved erratically during thunderstorms. He had tried to find the deflection of a compass needle when a current carrying wire was placed over the compass. But for a long time he had been unsuccessful, until one day in 1820 he placed the wire carrying an electric current parallel to the compass needle instead at right angles, as he had done previously.

Ampere's Current Force Law. This discovery set off a remarkable flurry of experimentation which produced immediately many discoveries. In fact, only a week after Oersted's discovery was reported to the French Academy of Sciences, Andre Ampere (1775-1836) presented the first of a series of papers on the magnetic effects of electric currents. To the science which deals with mutual action of currents he gave the name electrodynamics. In his early papers, Ampere showed that the currents in two parallel wires would attract one another if the current were in the same direction but would repel each other if the direction of the current in one of them was in the opposite direction to the current in the other wire. Jean-Baptiste Biot (1774-1862) and Felix Savart (1791-1841) made quantitative measurements of the forces between parallel currents and found that the force on the current in either wire was directly proportional to the common length of the wire and to the product of the currents in the wires and inversely proportional to the distance between the wires. That is,
F ∝ (I₁I₂L)/d,
or, expressed as a mathematical formula,
F = K(I₁I₂L)/d, (1)
where I₁ and I₂ are the electric currents, measured in amperes, flowing in the two parallel wires having a common length L between them, measured in meters, and are separated by the distance d, also measured in meters. The quantity K is the constant of proportionality and has a value of
2 × 10^-7 newtons/amp².
This constant is usually written as two times the coulomb constant of magnetism, k_m; that is,
K = 2k_m,
where k_m = 10^-7 nt/amp².
Thus Ampere's current force law is usually written
F = k_m(2I₁I₂L)/d. (2)
This relation is used to define the unit of current, the ampere.
Suppose that the wires are one meter apart (d = 1.0 m, exactly)
and that the two currents in them are equal (I₁ = I₂ = I).
If this common current is adjusted so that, by measurement, the force of attraction per unit length between the two wires is exactly 2 × 10^-7 newtons per meter, then the current is defined to be one ampere. By international agreement the ampere is the fourth fundamental unit (the other three are the kilogram, as the unit of mass, the meter as the unit of length, and the second as the unit of time). Since all electrical and magnetic units are expressed in terms of the coulomb as the unit of electric charge, the coulomb is defined as the quantity of charge that crosses an cross sectional area of an conductor when a current of one ampere passes through this area in one second.
That is, 1 coulomb = 1 ampere × 1 second or a coulomb is an amp-sec.

Biot-Savart Law. At a meeting of the Academy of Sciences on October 30, 1820, Biot and Savart announced that the action experienced by the north or south pole of magnet, when placed at any distance from a straight wire carrying an electric current, may be thus expressed: "Draw from the pole a perpendicular to the wire; the force on the pole is at right angles to this line and to the wire, and its intensity is proportional to the reciprocal of the distance." This result was soon further analyzed and the law known as the Biot-Savart Law was found that may be stated as follows: the magnetic force per unit pole due to an element ds of a circuit, in which a current I is flowing, at a point whose vector distance from ds is r, is in rationalized MKS (RMKS) system of units
dH = (1/4π)[I(ds × r₁)/r²], (3)
where r₁ = r/r is the unit vector in the direction from the current element Ids to the point where the magnetic force per unit pole is to be determined. Now H is the magnetic force per unit pole and is called magnetic field intensity; in the RMKS system of units it is measured in newtons per weber. The direction of H is the direction of the force exerted on an isolated point test north pole and is the direction of the vector or cross product of the current element and the unit vector as given by the right-hand rule.

Using this law, the results that Biot and Savart announced on October 30, 1820, can be found. Consider a long straight wire carrying a constant current I be along the x-axis of the rectangular coordinate system whose origin O is at the foot of the perpendicular R to the long straight wire from the point P where the magnetic field intensity H is to be determined. Let θ be the angle between r, the distance from the current element ds to the point P, and the direction of the current element along the x-axis. Using the scalar form the equation (3) and integrating, we get
H = ∫dH = (1/4π)(I)∫[(sin θ dx)/r²]. (4)
Let α be the complement of θ (α = 90° - θ), so that x = R tan α, the position of the current element on the x-axis, and R is the length of perpendicular distance from the long straight wire to the point P. Differentiating, we get
dx = R sec² α dα.
Also let r = R sec α, so that R/r = sin θ = cos α.
Integrating between -π/2 and π/2, we get
H = (1/4π)(I/R)[1-(-1)] = (1/4π)(2I/R). (5)
This is called the Law of Biot and Savart for the magnetic field intensity about a long straight current carrying conductor.

Using this law, Biot and Savart derived Ampere's current force law, equation (2) above, that is, the force per unit length between two long straight parallel conductors carrying currents I₁ and I₂ that are separated by a distance d is directly proportional to the common length of the wire and the product of the currents in the wires and inversely proportional to the distance between the wires. First, Biot and Savart restated Ampere's current force law in differential form.
The differential force dF exerted on the current element I₂ds in conductor two by the magnetic field of intensity H around conductor one is
dF = μI₂(ds × H), (6)
where μ is called absolute permeability and is equal to
μ = μ₀μ_r, (7)
where μ₀ is called the permeability of vacuum and is equal to 4π × 10^-7 newtons per amperes² and μ_r is called the magnetic permeability or relative permeability of the material in which the magnetic field is present and has a value of one or more depending upon the material
(μ_r = 1 for vacuum and nearly one for air).

This equation (6) is sometimes called the motor principle. Using the Law of Biot and Savart, equation (5), and taking the x-axis of our coordinate system in the direction of current element in conductor two, the magnitude of the differential force on conductor two, given by equation (6), becomes
dF₂ = μI₂(Hdx) = μI₂(2I₁/4πR)dx.
Integrating between -L/2 and L/2 and letting R be the distance d between the conductors, we get
F₂ = (μ/4π)[(2I₁I₂L)/d],
which is Ampere's Current Force Law, equation (2) above, where in vacuum or air
k_m = μ/4π = 10^-7 newtons/ampere². (8)

Magnetic Flux Density. After Michael Faraday introduced the concept of fields and lines of force (flux lines), the quantity called flux density was introduced. Magnetic flux density B at a point in a magnetic field was defined as the number lines of force (flux lines) per unit area of a surface perpendicular to the direction of the field at that point. That is,
B = dφ/dA, (9)
where φ is called magnetic flux or magnetic lines of force measured in webers and A is the area of a surface perpendicular to the direction of the field at the point where the flux density is to be determined and is measured webers per meter². The magnetic flux density at a point in the magnetic field is directly proportional to the magnetic field intensity at that point. That is,
B = μH, (10)
where μ is called absolute permeability and for vacuum
μ = μ₀ = 4π × 10^-7 newtons per amperes².

Magnetic field intensity H is measured in newtons per weber and is equal to ampere per meter, since the unit of flux is the weber that is defined as newtonmeter per ampere. That is,
mewton/weber = newton/(newton · meter/ampere) = ampere/meter.
Both magnetic flux density and magnetic field intensity are vector quantity; the direction of magnetic flux density is the same as the magnetic field intensity. Hence,
B = μH, and H = B/μ. (11)
Using this relation, the equations for magnetic field intensity H can be converted into equations for magnetic flux density B.
dB = (μ/4π)[I(ds × r₁)/r²]. (3A) Biot-Savart Law Equation.
B = (μ/4π)(2I/R). (5A) Biot-Savart Law for long straight wire.
dF = I₂(ds × B), (6A) The Motor Principle.
As an example of the application of the motor principle, let us compute the torque on a rectangular current circuit in a uniform magnetic field. Consider a rectangular coil suspended vertically at the middle of its top so that its plane is parallel to the magnetic flux density B. Let the length of the coil perpendicular to the field be L and its width W, and the current I in the coil. Using the equation (6A), let us compute the force on each side of the coil. At the top and bottom of the coil the force is zero, since ds and B then are parallel and ds × B = 0. On the side of the coil in which the current is flowing downward and which is perpendicular to B, the force per turn will be ILB normal to the plane of the coil. On the opposite side in which the current is flowing upward, the force will have same magnitude, but in the opposite direction. The result is a torque τ whose magnitude is give by
τ = NILBW = NIAB,
where A = LW is the area of the coil and N is the number of turns in the coil.

If the coil is allowed to turn, the torque due to the field will decrease and the restoring torque of the suspension fiber will increase unit equilibrium is reached. Letting be the angle turn trough and k be the stiffness constant for the fiber, we get
τ = NIAB cos θ = kθ.
If N, A and k are known and if θ is measured, we can calculate the magnetic flux density B in terms of the current I; that is,
B = kθ/(NIA cos θ).

According to equation (3A), dB is directly proportional to (ds × r₁)/r² and may be integrated over all the current elements of the entire circuit of which Ids is a part. Then, since the current I is constant, we can integrate equation (3A) and we get
B = (μ/4π)(I)∫_S[(ds × r₁)/r²] (12)
for the magnetic flux density due to a current circuit. The lines of flux representing B form closed circles about the conductor, as given by the following right-hand fist rule: If you imagine that the wire is grabbed by your right hand with your fingers curled about the wire and your thumb pointing in the direction of the positive current, then the fingers will curl in the direction of the lines of flux representing the magnetic field.

Ampere's Circuit Law. Ampere's Circuit Law is an alternative to the Biot-Savart Law equation (12) for electric currents and the magnetic field they generate. In some ways it is analogous to Gauss' Law for electrostatics, which is an alternate formulation of Coulomb's law. Ampere's Circuit Law is particularly useful for finding the magnetic field in situations of geometric symmetry. Consider equation (5A) that gives the magnetic flux density produced by an infinitely long straight conductor. Rearranging the equation, we get
B(2πR) = μI.
On the left side of this equation, the magnetic flux density B, at all points a distance R from the conductor, is multiplied by 2πR, the circumference of a circular loop at the distance R.
In general this expression can be found by the line integral of B around a closed path λ which we designate as
∫_λ(B · dλ) = B(2πR),
where dλ is an element of the path. This line integral implies that the components of B along the path element dλ are to be taken around a closed path. In the case of a circular loop about a long straight conductor, the direction of B will always coincide with the direction dλ, if the loop is traverse in the same direction as the magnetic lines of force. The current I is the current flowing in the conductor enclosed by the loop path. That is,
∫_λ(B · dλ) = μI, or, ∫_λ(H · dλ) = I, (13)
since by equation (11), H = B/μ. This equation (13), which is a mathematical statement of Ampere's Circuit Law, holds for any closed path about any configuration of electric conductors where the current I is taken to be the net current enclosed by the loop path. Let any surface S whose periphery is the path of integration λ. Then the sum of the currents linking the path is the total current passing through S. In terms of the current density J (the current per unit area), this total current through S is
I = ∫∫_S(J · dλ).
Hence, we get
∫_λ(B · dλ) = μ∫∫_S(J · dS), or
∫_λ(H · dλ) = ∫∫_S(J · dS). (14)
This is the usual mathematical form of Ampere's Circuit Law. Just as Gauss' Law enabled us to get quickly the expression for the electric field intensity in problems involving a high degree of symmetry, so Ampere's Circuit Law offers us, in some cases, the easiest method of finding the expression for B.

The Magnetic Field about a Linear Current Conductor. Ampere's Circuit Law gives a simpler method for obtaining equation (5A). Consider the magnetic field about a very long straight conductor. Let us find the B at some point P external to the conductor, that is a distance R from the conductor. In equation (13) let us take our path λ a circle of radius R from the conductor passing through P and with its center at the conductor. From symmetry consideration B must have the same magnitude at all points on this path. From Ampere's Circuit Law we know that B cannot be radial, but must be circular about the conductor, so that B and dλ are parallel. Hence,
∫_λ(B · dλ) = B∫_λ(dλ) = 2πRB.
Thus equation (13) becomes 2πRB = μI, or solving for B, we get
B = (μ/2π)(I/R),
which is equation (5A), Biot-Savart Law for long straight wire.

The Magnetic Field of a Toroidal Solenoid. Consider a doughnut-shaped coil of uniformly wound with wire carrying a current. Let there be N total turns, or N/L turns per unit length, where L is the mean or average circumference around the interior of the coil. In applying equation (13), let us take for the path λ to be this circular path of length L. From symmetrical considerations B must be constant and in the direction of the path. Hence
∫_λ(B · dλ) = B∫_λ(dλ) = BL.
Since the path λ threads the current N times, equation (13) becomes
BL = μNI,
or solving for B we get
B = μ(NI/L). (15)
As L increases, a toroidal solenoid becomes more and more similar to a very long straight solenoid. Hence equation (15) also gives B for a long straight solenoid with μNI/L ampere-turns per meter.

The Curl of B. From equation (14) we may easily obtain the expression for ∇ × B
(and ∇ × H).
Using Stokes' Theorem, which says that for any vector function A:
∫_λ(A · dλ) = ∫∫_S(∇ × A) · dS,
where S is a surface whose periphery is λ.
Thus equations (14) becomes
∫∫_S(∇ × B) · dS = ∫∫_S(μJ · dS), or, ∫∫_S(∇ × H) · dS = ∫∫_S(J · dS).
Since both surface integrals refer to the same surface, we may equate the integrands, and get
∇ × B = μJ, or,
∇ × H = J, (16)
which is the differential or point form of Ampere's Circuit Law.
Thus the curl of B is zero only where there is no current density, that is, if J = 0, then ∇ × B = 0.

If the current density J is due to the motion of a charge density ρ with a mean velocity v (all quantities refer to the immediate neighborhood of the point where ∇ × B or ∇ × H is being evaluated), then
J = ρv
and
∇ × B = μρv, or, ∇ × H = ρv, (17)
for a steady flow of charge.

The Divergence of B. Having adopted the Amperian approach to electromagnetism, it is assumed that all magnetic fields can be attributed to the motion of electric charges. The magnetic flux density B may be considered to be the sum or integral of fields as those given by equation by equation (3A); that is,
dB = (μI/4π)[(ds × r₁)/r²].
This equation tells us that the magnetic lines representing the direction of the field dB are circles about the axis of the current element Ids. These lines do not start at (diverge from) any point, nor do the stop at (converge toward) any point. Thus the field dB is source-free or solenoidal, that is its divergence is zero. Furthermore, the sum of any number of such fields will be solenoidal. That is,
∇ · B = 0. (18)
This equation is one of the fundamental equations of field theory and is one of Maxwell's Electromagnetic Equations. We shall now prove mathematically this equation.

Using the identity from Vector Analysis that says that for any two vectors such as a and b,
∇ · (a × b) = b · (∇ × a) - a · (∇ × b), (19)
let a = (μ/4π)Ids and
b = r₁/r², so that
a × b = [(μ/4π)Ids] × [r₁/r²] = [(μI)/4π][(ds × r₁)/r²] = dB,
which is Biot-Savart Law, equation (3A).
In equation (19) the ∇ is an operator involving space differentiation with respect to the coordinates of the point P where the field is being computed. As P moves, the distance r and the direction of unit vector r₁ may change but μ, I, and ds remain constant. Hence,
∇ × a = 0.
The vector function b = r₁/r² is in the radial direction with a magnitude inversely proportional to r². This is the same type of vector function as electric field intensity E, the electric field due to a point charge. In a similar way as ∇ × E was zero,
∇ × b = 0.
Thus the right side of equation (19) is zero, so that the equation (19) reduces to
∇ · (a × b) = 0.
Since a × b = dB, then ∇ · dB = 0.
And since
∇ · B = ∇ · ∫_S(dB) = ∫_S(∇ · dB,
and ∇ · dB = 0, it follows that
∇ · B = 0, which is equation (18),
Maxwell's Second Equation of Electromagnetics.
Comparing this with the electrostatic equation ∇ · D = ρ, we see that the attempt to find the source of magnetic fields in magnetic "poles," analogous to isolated electric charges, the volume density of such poles must be everywhere zero. This older approach interpreted this as meaning that north and south poles always appear in pairs. The Amperian approach interprets this as meaning that the concept of poles are unnecessary and need not be introduced as the source of magnetic fields.

The Vector Potential. In Vector Analysis it was shown that a vector function whose divergence is always zero can be expressed as the curl of another vector function. That is,
if ∇ · B = 0, then B = ∇ × A, (20)
where A is called the vector potential. We shall now derive an expression for A at the point P, given the current I in the circuit produces the magnetic field B. First, let us show that
if ds has the components ds_x, ds_y, and ds_z, and r = xi + yj + zk and
r = (x² + y² + z²)^½, then
∇ × (ds/r) = (ds × r₁)/r². (21)
Using the definition of the curl,
∇ × A = i[∂A_z/∂y - A_y/∂z] + j[∂A_x/∂z - ∂A_z/∂x] + k[∂A_y/∂x - ∂A_x/∂y],
substituting (ds/r) for A, we get
∇ × (ds/r) =
i[∂(ds_z/r)/∂y - ∂(ds_y/r)/∂z] + j[∂(ds_x/r)/∂z - ∂(ds_z/r)/∂x] + k[∂(ds_y/r)/∂x - ∂(ds_x/r)/∂y]
∇ × (ds/r) = i[-(y/r³)ds_z + (z/r³)ds_y] + j[-(z/r³)ds_x + (x/r³)ds_z] + k[-(x/r³)ds_y + (y/r³)ds_x]
∇ × (ds/r) = (1/r³)[i(yds_z - zds_y) + j(zds_x - xds_z) + k(xds_y - yds_x)]
∇ × (ds/r) = (1/r³)(r × ds) = (1/r³)(ds × r) = ds × (1/r³)(r) = ds × (1/r²)(r/r) = (ds × r₁)/r², or,
∇ × (ds/r) = (ds × r₁)/r²,
where r₁ = r/r is the unit vector in the direction of r. As in a previous section, ∇ operates differentially upon r, but not on ds. Now substituting this into equation (12), we get
B = (μ/4π)∫_S[∇ × (ds/r)].
Since the differentiation involved in ∇ and the integration around the circuit are independent of each other, then the curl can be taken outside of the integral sign and we get
B = ∇ × A, where
A = (μ/4π)(I)∫_S(ds/r) (22)
is the expression for the vector potential that we were seeking.

The vector potential A given by equation (22) is not the only one whose curl is equal to B. If another vector function A′ is added to A, such as A′ = ∇Φ, where Φ is a scalar function, then we find that
∇ × (A + A′) = ∇ × A + ∇ × A′ = ∇ × A + ∇ × ∇Φ = ∇ × A = B,
since the curl of the gradient is always zero. Therefore, the potential function A must not be considered to be completely determined by equation (22).

The vector potential may also be expressed in terms of the current density J, or the product of the volume density ρ of charge and the velocity v of the moving free charges producing the magnetic field at point P. Suppose that the current element Ids has a cross section area A and volume dV = Ads. Then
Ids = JAds = JAds = JdV,
since ds is in the direction of J. Hence, equation (22) can be written as
A = (μ/4π)∫∫∫(JdV/r), (23)
where the volume integral is taken over all regions containing a current density ρ. Since J = ρv, then the equation (23) may be written as
A = (μ/4π)∫∫∫(ρv/r)dV. (24)
While ρ and v may vary throughout space, they must be subject to the restriction that they are independent of time, so that the magnetic field will be steady. In the case of a single charge, the magnetic field that it produces at a point P will vary as the charge moves along; such a field is not steady. For a charge q moving with the constant velocity v, we may put qv for Ids or ρvdV, provided that the charge is moving with a speed considerably less than the speed of light c, so that relativity effects may be neglected. Let us find the magnetic flux density B from the vector potential A. Using equation (24), we get
A = (μ/4π)(qv/r). (25)
Let us take the curl of this vector potential A, remembering the ∇ operates differentially only on the 1/r factor, since q and v are independent of the coordinates. Then we get
B = ∇ × A = (μq/4π)[∇ × (v/r) = (μq/4π)[(v × r₁)/r²], (26)
since by the same procedure that obtained equation (21), we get
∇ × (dv/r) = (dv × r₁)/r².
Equation (26) gives us the magnetic flux density due to a slowly moving charge. This is the same result that we get when we substitute qv for Ids in equation (12).

Force Exerted on a Moving Charge. At the end of the nineteenth century the Dutch physicist Hendrik Antoon Lorentz (1853-1928) in 1892 published his first memoir in which proposed a theory that all electrodynamic phenomena were ascribed to the agency of moving electric charges. This theory of Lorentz, like those of Weber, Riemann and Clausius, was a theory of electrons, which were supposed in a magnetic field to experience forces that proportional to their velocities, and to communicate these forces to the ponderable matter with which they might be associated. His theory is based on the four equations of Maxwell to which he added the equation which determines the ponderomotive force on a charged particle. This equation is now known as the Lorentz Force Law. This equation was first proposed by Oliver Heaviside (1850-1925) in his 1889 paper on moving charges in the Philosophical Magazine. Heaviside gave for the first time the equation for a mechanical force exerted on an electric charge which is moving in a magnetic field. That is, the force is equal to the charge multiplied by the vector product of the velocity v of the charge and magnetic flux density B.
F = q(v × B). (27)
This equation can be derived from the Motor Principle, equation (6A), where the current element Ids is replaced with vdq. Assume that the current is constant and in the current element there is a relative motion of a certain amount of charge, q. Also assume that charge has a constant forward velocity v and moves a displacement ds in the time interval dt. Then,
Ids = (dq/dt)ds = (ds/dt)dq = vdq, or
Ids = vdq.
Substituting vdq for Ids in the Motor Principle, equation (6A), we get
dF = I(ds × B) = Ids × B = vdq × B
dF = dq(v × B).
Integrating both sides of this differential equation between the limits from 0 to F, and from 0 to q.
F = q(v × B).
Lorentz added to this force exerted by the magnetic field the force exerted by an electric field.
F = qE + q(v × B) = q(E + v × B). (28)
To illustrate Lorentz Force Law consider the magnetic deflection of a beam of charged particles. Suppose that we have a beam of particles, each with a mass m, a charge e, and a velocity v. Let the beam move through a region where the magnetic flux density has a uniform value B. Using the equation (27) the force on each particle is
F = ev × B. (29)
Let B₌ and B_⊥ be the components of B parallel and perpendicular to v, respectively. The B₌, being parallel to the direction of the velocity, will not contribute to the force and may be ignored. But since B_⊥ is perpendicular to the direction of the velocity, a force of evB will act at right angles to the direction of motion and to that of B. As a result, the motion in the direction of B will remain unchanged, but the vector v will rotate in a cone about the direction of B. In other words, the path of the beam will be a helix (spiral) with its axis parallel to B. Charged particles traveling in the earth's magnetic field move in just such paths.

When the velocity v is perpendicular to B, the force F = evB. Since the force is at right angles to v, the motion is circular. From Newton's Second Law of Motion (F = ma) and the equation for centripetal acceleration (a = mv²/R), we get
evB = (mv²)/R,
or
(e/m) = (v/BR). (30)
This is a very useful equation when the mass or speed of an atomic or subatomic particle needs to be determined.

Magnetic Field of a Moving Charge. According to Lorentz Force Law, the electric charge must be in motion in order that a force exerted on it by a magnetic field; if the charge is at rest (v = 0), then there will be no force exerted on the charge by the magnetic field. But it is also true that the source of magnetic field must also be charges in motion. Experiments have shown that electric currents, or moving charges are needed to produce a magnetic field. This fact is expressed by the following equation.
B = (1/c²)(v × E), (31)
where B = magnetic flux density at a point r meters from q,
E = electric field intensity produced by q at a point r meters from q,
v = velocity of q with respect to some observer, and c = the speed of light.
We will now derive this equation.
Substituting vdq for Ids in Biot-Savart Law, equation (3A), we get
dB = (μ/4π) [(v × r₁)/r²]dq.
Integrating both sides of this differential equation between the limits from 0 to B, and from 0 to q.
B = (μq/4π) [(v × r₁)/r²], (32)
where B = magnetic flux density at a point r meters from q,
q = moving source charge,
v = velocity of q₁ with respect to some observer.
Let μ = 1/(εc²) = 4π × 10^-7 (nt/amp²), where c = the speed of light, then
B = (1/εc²)(q/4π) [(v × r₁)/r²] = (1/c²)(v × [q/(4πεr²)]r₁. (33)
Now the electric field intensity E at any point about an isolated point source q is given by the equation.
E = [q/(4πεr²)]r₁.
Hence, substituting this into equation (33) we get the equation
B = (1/c²)(v × E),
which is equation (31) that we wanted to derive.

Magnetic Flux. As electric fields are pictured by drawing lines of force to show the direction and to represent the strength of the field by the concentration of these lines, the same can be done for magnetic fields. This is done by defining the total magnetic flux φ crossing a surface S as
φ = ∫∫_S(B · dS), (34)
where B is the magnetic flux density that is defined as the number of lines of force or flux per unit area through a surface S whose normal is in the direction of B and dS is an element of the surface S.

Gauss' Law for Magnetism. When the surface S is a closed surface, by convention the vector dS is taken as positive when it points toward outside of the closed surface. Then, the magnetic flux is positive when the magnetic line pass outward through the surface and is negative when the lines pass inward. If the total magnetic flux through a close surface is zero, then this implies that an equal number of magnetic lines enter and leave the closed surface. This is precisely what is found experimentally and is expressed in the following equation that is called Gauss' Law for Magnetism.
φ = ∫∫_S(B · dS) = 0. (35)
This implies that magnetic lines of force do not terminate on any thing but its own "tail" and that magnetic lines of force are always closed loops. This also implies that there are no isolated magnetic poles in the same way that electric lines of force terminate on electric charges. On a permanent magnet the magnetic lines of force do not originate on the north pole and terminate on the south pole; the lines of force passes through the interior of the magnet and exits at the north pole and enters the magnet at the south pole, forming closed loops. Neither do monopoles exist in the physical world, as the most subtle experiments confirm.