doPhysics · Electromagnetism 2
Sources of the Magnetic Field
Where a magnetic field comes from. The Biot-Savart law gives the field of every small piece of a current, and adding the pieces gives straight wires, arcs, loops and polygons. Ampere's circuital law gives the field of symmetric currents in one line, inside cylinders and cables, solenoids and toroids.
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Why this chapter
Why this chapter exists
The last two chapters found what a magnetic field does to charges and currents, but always took the field as given. This chapter finds where it comes from: every current makes one. A single law gives the field of each small piece of a wire, and adding the pieces gives the field of any shape of wire. For currents with strong symmetry a second law does the same work in one line. Together they explain a compass that jumps beside a starter cable, a quiet coaxial cable and the huge field inside an MRI scanner.
You will learn
By the end of this chapter you will
- State the right-hand thumb rule and draw the field lines of a straight current.
- State the Biot-Savart law and use it to find the field of a long straight wire, \(B = \mu_0 I/2\pi d\).
- Find the field of a finite straight wire from the angles to its ends. Include the end of a long wire and points on its line.
- Find the field at the centre of a loop, a coil and an arc, and on the axis of a loop. Add the fields of polygons and pairs of loops.
- State Ampere's circuital law with its sign rule, and use it for long wires, solid and hollow cylinders and coaxial cables.
- Derive \(B = \mu_0 nI\) inside a long solenoid, and describe its field at the ends, outside and with an iron core.
- Find the field of a toroid, and decide when Ampere's law can give a field and when the Biot-Savart law is needed.
Before you begin
Before you begin
From the chapter Magnetic Force on Moving Charges you should know the magnetic field \(B\), its unit the tesla, and the force \(qv \times B\). From the chapter Magnetic Force on Currents and the Magnetic Dipole you should know that a current loop is a magnetic dipole of moment \(IA\). You should be able to take a cross product with the right hand, as for torque, and integrate \(\cos\phi\) and powers of \(r\).
Lessons
1
Oersted's Compass and the Biot-Savart Law
A current makes a magnetic field. Its lines circle the wire, the right hand gives their sense, and the Biot-Savart law gives the field of each small piece of wire. Adding the pieces of a long straight wire gives \(B = \mu_0 I/2\pi d\).
A current makes a fieldThe biot savart lawField of a long straight wire
2
Finite Wires and the Point on the Line
A straight wire of any length gives \(B = (\mu_0 I/4\pi d)(\sin\theta_1 + \sin\theta_2)\), with the angles to its two ends. At the end of a long wire the field is half, on the line of a wire it is zero, and fields of several wires add as vectors.
Field of a finite straight wireAt the end of a long wireZero on the line of the current
3
Loops and Arcs
At the centre of a circular loop every element pushes the field the same way, so \(B = \mu_0 NI/2R\), and an arc gives its share \(\theta/2\pi\). On the axis the field falls as \(1/x^3\), like a dipole's. Squares, polygons and pairs of loops are sums of pieces already found.
Field at the centre of a loopField of an arcOn the axis of a loopSuperposition of loops and polygons
4
Ampere's Circuital Law
Go once round any closed loop and add \(B \cdot dl\). The total is \(\mu_0\) times the current that threads the loop, whatever the shape of the loop. Where symmetry makes \(B\) constant along a loop, this gives the field in one line, inside and outside wires, cylinders and cables.
Amperes circuital lawThe long wire by amperes lawInside and outside a solid cylinderHollow cylinders and the coaxial cable
5
The Solenoid
A long coil of many turns holds a strong, even field inside, \(B = \mu_0 nI\), set by the turns per metre and the current and not by the radius. At an end the field is half, outside it is nearly zero, and an iron core multiplies it hundreds of times.
Field inside a long solenoidEnds and outside of a solenoidThe iron core and the mri magnet
6
The Toroid, and Where Ampere's Law Fails
A solenoid bent into a ring keeps its whole field inside the ring, \(B = \mu_0 NI/2\pi r\), and none in the hole or outside. Ampere's law is always true, but it finds \(B\) only where symmetry allows; a finite wire or a single loop needs the Biot-Savart law.
The toroidWhen amperes law cannot be usedChoosing between biot savart and ampere
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- Amperian loop
- A closed path chosen for Ampere's circuital law, usually one on which symmetry makes the field constant and along the path.
- Biot-Savart law
- The field of a current element: \(d\vec B = (\mu_0/4\pi)\,I\,d\vec l\times\hat r/r^2\), of size \((\mu_0/4\pi)\,I\,dl\sin\theta/r^2\).
- coaxial cable
- A cable with a central core and a metal shell round it, carrying a current out along one and back along the other. Its magnetic field is confined to the inside.
- current density
- The current per unit area of a conductor's cross-section, \(J\), in A/m². A uniform current \(I\) in a wire of radius \(R\) has \(J = I/\pi R^2\).
- current element
- A very short piece of a current-carrying wire, written I dl, with the direction of the current. The field of a wire is the sum of the fields of its elements.
- enclosed current
- The net current threading an Amperian loop, each current counted positive or negative by the right-hand rule with the loop's sense.
- field line
- A line drawn in a field so that the field at every point is along it. The field lines of a straight current are circles round the wire.
- Helmholtz coils
- Two equal coaxial coils one radius apart, carrying the same current the same way. The field between them is nearly uniform over a wide region.
- line integral
- The sum of \(\vec B\cdot d\vec l = B\,dl\cos\theta\) over small steps along a path. Round a closed path it is written \(\oint\vec B\cdot d\vec l\).
- magnetic dipole
- A current loop, or a small magnet, seen from far away. A loop of moment \(M = IA\) has an axial field \((\mu_0/4\pi)(2M/x^3)\) at a large distance \(x\).
- permeability of free space
- The constant \(\mu_0 = 4\pi\times10^{-7}\,\text{T m/A}\) in the laws for magnetic fields of currents in vacuum or air; \(\mu_0/4\pi = 10^{-7}\,\text{T m/A}\).
- relative permeability
- The factor \(\mu_r\) by which a material filling a solenoid multiplies its field. Soft iron has \(\mu_r\) of several hundred or more.
- right-hand thumb rule
- Grip a straight wire with the right hand, thumb along the current; the curled fingers give the sense of the field lines round it.
- semi-infinite wire
- A straight wire that starts at a point and runs a very long way in one direction. On the perpendicular through its end its field is \(\mu_0 I/4\pi d\), half that of an endless wire.
- solenoid
- A long coil of wire wound in a tight helix. Inside a long solenoid the field is uniform, \(B = \mu_0 nI\).
- superposition
- The rule that the field of several currents, or of several pieces of one wire, is the vector sum of their separate fields.
- toroid
- A coil wound on a ring, a solenoid bent round until its ends meet. Its field, \(\mu_0 NI/2\pi r\), lies wholly inside the ring.
- turns per unit length
- The number of turns of a solenoid per metre of its length, \(n = N/\ell\).