Magnetic fields

Currents, i.e. moving electric charges, produce magnetic fields.  There are no magnetic charges.   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.

(1) A E∙dA = Qinside0
(2) Γ E∙ds = 0
(3) A B∙dA = 0
(4) Γ B∙ds =  μ0Ithrough Γ

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 =  μ0Ithrough 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.


imageAmpere'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 =  μ0Ithrough Γ.
B = μ0I/(2πr).

imageThe 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.


The force between two parallel wires

imageParallel wires carrying currents will exert forces on each other.  Each wire produces a magnetic field, which influences the other wire.  When the currents in both wires flow in the same direction, then the force is attractive.  When the currents flow in opposite directions, then the force is repulsive.

In the diagram on the right, both wires carry a current in the same direction.  The magnetic field produced by wire 1 at the location of wire 2 is B = μ0I1/(2πd) pointing into the page.  The force on a section of wire 2 of length L is is F = I2L × B.  F  has magnitude F = I2LB = (μ0I1I2L)/(2πd) and points towards wire 1.  The force per unit length is (F/L) = (μ0I1I2)/(2πd) towards wire 1.  Similarly the force per unit length on wire 1 due to the field produced by wire 2 is (F/L) = (μ0I1I2)/(2πd) towards wire 2.  (Newton's third law)

Two parallel wires carrying current in the same direction  attract each other.
Two parallel wires carrying current in opposite direction  repel each other.

Link:  Two wires  (Youtube)

imageProblem:

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:
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
B = 4μ0I sin(45o)/(2πd) = 80*π*10-7sin(45o)/(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) = km[I dl × (r-r')/|r-r'|3.

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

imageThe 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

imageThe 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 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.)

imageThe 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.

image

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:
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.

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.

image

Solution:
We use the principle of superposition and add the magnetic field of a long straight wire and a current loop.
Bwire(r) = μ0I/(2πR) out of the page.
Bloop(r) = μ0I/(2R)]  out of the page.
B
total(r) = [μ0I/(2πR)][π + 1]  out of the page.