Electric charges produce electric fields. 
The electric field produced by a point charge q at rest at the origin is E = F/q = (k_eQ/r²) (r/r).
The electric field of a charge distribution can be found using the principle of superposition.  Field lines can be used to visualize the electric field.

There are no magnetic charges.

Moving electric charges produce magnetic fields. 
To produce a magnetic field B, a current density j is needed, i.e. j = ρ₊<v₊> + ρ_-<v_-> cannot be zero. 

The magnetic field produced by permanent magnets is due to "magnetization current" densities inside ferromagnetic materials.  This is atomic-scale physics, and on that scale Classical Physics fails and Quantum Mechanics is the model that is used.  Many research groups are currently investigating the magnetic properties of various materials, both experimentally and theoretically, and many questions still need to be answered.

Field lines can be used to visualize the magnetic field.
Magnetic field lines have no beginning or end, they always form closed loops.  (Because there are no magnetic charges, there are no sources or sinks.)
The field lines visualizing the magnetic field of a permanent bar magnet are shown on the right.

External link:  Explore the magnetic field of a bar magnet (PhET)

We define the north and south pole of a bar magnet so that inside the bar magnet the magnetic field points from the south pole (S) to the north pole (N).  Outside a bar magnet the field lines close the loop, from north to south.  The density of the field lines is proportional to the strength of the magnetic field.

When a bar magnet is bend into the shape of a horseshoe, the magnetic field between the poles is nearly uniform and usually quite strong.

Currents, i.e. moving electric charges, produce magnetic fields.  Maxwell's equations tell us how to compute the electric fields and magnetic fields produced by charged particles.  The terms electrostatics and magnetostatics refer to steady state conditions, when all charges are at rest or only steady currents are flowing.  Then the charge densities do not change anywhere.  Under those conditions Maxwell's equations are given on the below.

The first of these equations is Gauss' law.  It tells us that the total flux of the electric field through a closed surface is proportional to the total charge inside the surface.  In electrostatics, electric field lines begin and end on charges.  If there is no charge inside a closed surface, then there is no source or sink of field lines inside the surface.  All the field lines that enter through the surface into the volume enclosed by the surface must also exit through the surface.  The net flux through the surface is zero.
Gauss' law by itself can be used to find the electric field of a point charge at rest, and the principle of superposition can then be used to find the electric field of an arbitrary charge distribution.

The second of Maxwell's equations tells us that the electrostatic field is a conservative field.  The total work done by the electric field on a test charge when moving it along a closed loop path is

W = q∮_Γ E∙ds = 0.

Under steady state conditions, Maxwell's equations for the electric field are decoupled from the equations for the magnetic field.  The first two of Maxwell's equations describe the properties of the electric field, while the last two describe the properties of the magnetic field.  We can calculate the electric field without considering the magnetic field and vice versa.

The third of Maxwell's equations tells us that there are no magnetic charges, and therefore no sources and sinks for the magnetic field.  The net flux of the magnetic field through any closed surface is zero.  All the field lines that enter through the surface into a volume enclosed by the surface also exit through the surface.  Magnetic field lines always are closed loops.

The fourth of Maxwell's equations is called Ampere's law.  It tells us that the circulation of the magnetic field B, namely the integral of the tangential component of B along a closed curve Γ, is proportional to the current flowing through the area enclosed by the curve.  The proportional constant μ₀ = 4π10^-7 N/A² is called the permeability of free space.

∮_Γ B∙ds =  μ₀I_{through Γ}

When integrating along a closed path, one must choose a direction for ds, i.e. one must choose to move around the loop clockwise or counterclockwise.  The two directions yield the same magnitude but a different sign for the result of the integration.  The right-hand rule is again used to resolve this ambiguity.  Let your thumb point in the direction of current flow.  The circulation of B is positive if the fingers of your right hand point in the direction of ds.

Ampere's law can be used to find the magnetic field due to a steady current flowing in a very long straight wire.  Consider a single, isolated, long straight wire, carrying a current I.  Assume the wire is aligned with the z-axis.  Symmetry considerations tell us that the magnitude of the magnetic field produced by the current in the wire at some point P(x,y,z) can only depend on the radial distance r = (x² + y²)^½ of P away from the wire, not on the z-coordinate of P.  If we imagine a circular loop of radius r, then the magnetic field is tangential to the loop.  The integral of the tangential component of B along the closed curve is just the magnitude of B times the circumference of the loop.

∮_Γ B∙ds = B2πr =  μ₀I_{through Γ}.
B = μ₀I/(2πr).

The proportional constant μ₀ = 4π*10^-7 N/A² is called the permeability of free space.
The SI unit for the magnetic field is Ns/(Cm) = Tesla(T)

The magnetic field produced by a steady current flowing in a very long straight wire encircles the wire.   At a point P a radial distance r away from the wire it has magnitude B = μ₀I/(2πr).  The direction of B is given by the right-hand rule.  Let your thumb point into the direction of the current flow.  Your fingers curl into the direction of the magnetic field produced by the current.  The picture shows an iron-filing pattern, which reveals the nature of the magnetic field surrounding a current-carrying wire.

Problem:

A 30 A current is flowing in a long, straight wire.  What is the magnetic field strength a distance 1 cm from the wire?

Solution:

• Reasoning:
  The magnitude of the magnetic field field a radial distance r away from a long, straight wire is B = μ₀I/(2πr).
• Details of the calculation:
  B = (4π*10^-7 N/A²)*30 A/(2π*0.01 m) =  1.2*10^-5/*0.02 = N/(As) = 6*10^-4 T.
  For comparison, near Knoxville, TN, the strength of the Earth magnetic field is ~ 53 microT = 5.3*10^-5 T.

Problem:

Four long parallel wires carry equal currents of I = 5 A. The figure on the right is an end view of the conductors.
The current direction is into the page at points A and B and out of the page at points C and D.  Calculate the magnitude and direction of the magnetic field at point P, located at the center of the square of edge length 0.2 m.

Solution:

• Reasoning:
  At point P, (the origin of the chosen coordinate system) wires A and D each produce a magnetic field of magnitude μ₀I/(2πd) pointing towards wire B.  Here d² = (0.1² + 0.1²) m².  Wires C and B each produce a magnetic field of magnitude μ₀I/(2πd) pointing towards wire D.  The y-components of all the fields add, while the x-components cancel.  The total field at point P therefore has magnitude
• Details of the calculation:
  B = 4μ₀I sin(45^o)/(2πd) = 80*π*10^-7sin(45^o)/(2π(0.2)^½) T = 20 mT and points into the negative y-direction.

Maxwell's equations can be used to derive the Biot-Savart law.  The Biot-Savart law can be used to find the magnetic field produced by any distribution of steady currents.  For a small segment of wire of length dl carrying a current I, the Biot-Savart law states that the magnetic field dB produced by that segment a distance r from the segment is given by

dB(r) = k_m I dl ×(r-r')/|r-r'|³.

Here k_m is a constant, k_m = 10^-7 Tm/A.  Often k_m is written as k_m = μ₀/(4π), where μ₀ is the permeability of free space.

The Biot-Savart law is an inverse square law.
(r-r')/|r-r'|³ = 1/|r-r'|² * (r-r')/|r-r'| where (r-r')/|r-r'| is the unit vector pointing in the (r-r') direction.
The directional aspects are such that the magnetic field produced by a current element dl at any point P encircles the straight line passing through dl.

Since steady currents always flow in closed loops, we need to integrate dB over the entire circuit to evaluate the net field B at point P.  (This is a vector integral.  The contributions dB from different sections add vectorially.)  All sections of the loop contribute to B.  But because of the inverse-square dependence on distance, the sections closest to P make the largest contributions.

B(r) = (μ₀/(4π))∫_circuit I dl'×(r-r')/|r-r'|³.

Some Results

The magnetic field on the axis of a current loop of radius R, a distance z from the center of the loop is

B = μ₀IR²/(R² + z²)^3/2 n.

Here n is a unit vector pointing in the direction of the right-hand rule, if the fingers coil in the direction of the current flow.
At the center of the loop z = 0 and B = (μ₀I/(2R))n.

A small current loop has a magnetic dipole moment defined as μ = IAn.  The magnetic field lines of a loop with dipole moment μ are shown on the right.  The field-line pattern is that of a small magnet.  We say that μ points from the south pole to the north pole of the magnet.  (Magnetic field lines therefore exit a magnet at the north pole and enter at the south pole.)

The magnitude of the field inside a tightly wound solenoid (away from the ends) with n = N/L turns per unit length is

B = μ₀nI.

The direction of the field is given by the right-hand rule.  Curl the fingers of your right hand in the direction that the current flows in the solenoid.  Your thumb points in the direction of B. In the picture on the right the field pattern inside a current-carrying solenoid is revealed with iron filings.

Problem:

What current is required in the windings of a long solenoid that has 1000 turns uniformly distributed over a length of 0.4 m in order to produce at the center of the solenoid a magnetic field of 10^-4 T?

Solution:

• Reasoning:
  The magnitude of the field inside a tightly wound solenoid (away from the ends) with n = N/L turns per unit length is B = μ₀nI.
• Details of the calculation:
  Near the center of the solenoid B = μ₀nI = 4π10^-7(1000/0.4)*I T, with I in units of Ampere.
  With B = 10^-4 T, we need I = 10^-4A/(4π*10^-7 2500) A = 31.8 mA.

External link:  Explore the magnetic field produced by a coil (PhET)

Problem:

Find the magnetic field at the center of a circular loop of radius R that is formed in a long straight thin wire that carries current I.  Use the SI system of units.

Solution:

• Reasoning:
  We use the principle of superposition and add the magnetic field of a long straight wire and a current loop.
• Details of the calculation:
  B_wire(r) = μ₀I/(2πR) out of the page.
  B_loop(r) = μ₀I/(2R)]  out of the page.
  B_total(r) = [μ₀I/(2πR)][π + 1]  out of the page.

The right hand rule

We use a right hand rule for two different purposes.

One is to find the direction of the vector cross product.  That is covered in module 4.

The other purpose is to relate the direction of a current to the direction of the magnetic field or to the magnetic moment it produces.  There are two cases where this works.

(a)  We have current flowing in a straight wire and the magnetic field lines form circles around the wire.
(b)  We have current flowing in a wire forming a loop or a coil, and the magnetic field in the middle of the loop or coil is perpendicular to the loop and straight inside the coil.

In other words, we have something that is straight, and something that curls.
Then if the thumb of your right hand points in the direction of the thing that is straight, then the fingers of you right hand curl in the same direction of the thing that is curling, and vice versa.