Sources of Magnetic Fields

Sources of Magnetic FieldsLevel 5 Physics

January 2013

Material adapted from MIT 8.02 course notes

Sources of Magnetic Fields

Introduction

Observation

Magnets

Where do you think magnetic fields originate from?

A magnet is perhaps the most commonly known source of magneticfields, but the actual cause of the magnetic field is not obvious.

An interesting feature of magnetism is that the magnetic polesalways come in a pair. The north pole and south pole of a magnetcannot be separated to create magnetic “monopoles”.

Sources of Magnetic Fields

Introduction

Observation

Magnets

Where do you think magnetic fields originate from?

A magnet is perhaps the most commonly known source of magneticfields, but the actual cause of the magnetic field is not obvious.

An interesting feature of magnetism is that the magnetic polesalways come in a pair. The north pole and south pole of a magnetcannot be separated to create magnetic “monopoles”.

Sources of Magnetic Fields

Introduction

Observation

Magnets

Where do you think magnetic fields originate from?

A magnet is perhaps the most commonly known source of magneticfields, but the actual cause of the magnetic field is not obvious.

An interesting feature of magnetism is that the magnetic polesalways come in a pair. The north pole and south pole of a magnetcannot be separated to create magnetic “monopoles”.

Sources of Magnetic Fields

Introduction

Observation

Magnetic Field Observations

Some observations we have already made and formulas we havederived give us clues for the source of magnetic fields:

Unlike poles attract while like poles repel

Magnetic force on a moving point charge: ~FB = q~v × ~BMagnetic force on a current-carrying wire: ~FB = I ~̀× ~B

Any ideas?The motion of charges causes magnetic fields.

Sources of Magnetic Fields

Introduction

Observation

Magnetic Field Observations

Some observations we have already made and formulas we havederived give us clues for the source of magnetic fields:

Unlike poles attract while like poles repel

Magnetic force on a moving point charge: ~FB = q~v × ~BMagnetic force on a current-carrying wire: ~FB = I ~̀× ~B

Any ideas?The motion of charges causes magnetic fields.

Sources of Magnetic Fields

Introduction

Observation

Magnetic Field Observations

Some observations we have already made and formulas we havederived give us clues for the source of magnetic fields:

Unlike poles attract while like poles repel

Magnetic force on a moving point charge: ~FB = q~v × ~BMagnetic force on a current-carrying wire: ~FB = I ~̀× ~B

Any ideas?

The motion of charges causes magnetic fields.

Sources of Magnetic Fields

Introduction

Observation

Magnetic Field Observations

Some observations we have already made and formulas we havederived give us clues for the source of magnetic fields:

Unlike poles attract while like poles repel

Magnetic force on a moving point charge: ~FB = q~v × ~BMagnetic force on a current-carrying wire: ~FB = I ~̀× ~B

Any ideas?The motion of charges causes magnetic fields.

Sources of Magnetic Fields

Introduction

Ørsted

Ørsted’s Discovery

In 1820, Ørsted noticed that a wire carrying an electric currentcaused a compass needle to deflect.

He was one of the first to discover aconnection between electricity andmagnetism, and he eventuallydetermined that an electric currentproduces a circular magnetic field.

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Straight Wire Magnetic Field

Remember that iron filings align towards the magnetic field lines.

If we apply a current along a straight wire, the alignment of theiron filings indicates a circular magnetic field.

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Straight Wire Magnetic Field

Remember that iron filings align towards the magnetic field lines.

If we apply a current along a straight wire, the alignment of theiron filings indicates a circular magnetic field.

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Semicircle Wire Magnetic Field

The direction of the magnetic field in the center of a semicirclewire consists of parallel lines.

This suggests that the direction of the current affects the directionof the magnetic field (otherwise we might expect the magneticfield to have zero magnitude in the center).

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Another Right Hand Rule

The magnetic field produced by a current-carrying wire is circular,with the direction given by another right hand rule.

Right Hand Rule

If your thumb points along the direction of the current, yourfingers curl in the direction of the magnetic field.

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Solenoid

A solenoid is a piece of wire wrapped into a coil. An ideal solenoidhas infinite length and the coil is tightly packed.

What do you think the magnetic field for a solenoid looks like,considering the magnetic field for a straight wire and the righthand rule?

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Solenoid

A solenoid is a piece of wire wrapped into a coil. An ideal solenoidhas infinite length and the coil is tightly packed.

What do you think the magnetic field for a solenoid looks like,considering the magnetic field for a straight wire and the righthand rule?

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Solenoid Magnetic Field

The magnetic field is strong inside the solenoid and consists ofparallel lines with direction determined by the direction of thecurrent.

The magnetic field is weak outside the solenoid and for an idealsolenoid is zero.

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Toroid

A toroid is a solenoid wrapped around into a circle.

What do you think the magnetic field for a toroid looks like,considering the magnetic field for a solenoid?

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Toroid

A toroid is a solenoid wrapped around into a circle.

What do you think the magnetic field for a toroid looks like,considering the magnetic field for a solenoid?

Sources of Magnetic Fields

Introduction

Magnetic Field Lines

Toroid Magnetic Field

The magnetic field lines for a toroid are circular in the region insidethe loops.

Outside the toroid, the magnetic field is zero.

Sources of Magnetic Fields

Theory

Biot-Savart

Biot-Savart Law

Biot-Savart Law

The magnetic field contribution, d~B,from a current source, Id~s, at a fieldpoint P a distance r in the direction ofr̂ is given by

d~B =µ0

4π

Id~s× r̂

r2

where µ0 is the permeability of freespace constant with value

µ0 = 4π × 10−7T ·m/A

Sources of Magnetic Fields

Theory

Biot-Savart

Summing the Contributions

If the differential contribution is given by

d~B =µ0

4π

Id~s× r̂

r2

then taking the integral over the length of the wire gives anequation for calculating the magnetic field:

~B =

∫wire

d~B =µ0I

4π

∫wire

d~s× r̂

r2

Remember that an integral represents a sum where the d~s are verysmall.

Sources of Magnetic Fields

Theory

Biot-Savart

Finite Length Straight Wire

The magnetic field caused at point P by a finite length straightwire can be calculated using Biot-Savart’s Law.

Performing the calculation gives for the magnitude

B =µ0I

4πa(cos θ2 + cos θ1)

Sources of Magnetic Fields

Theory

Biot-Savart

Finite Length Straight Wire

The magnetic field caused at point P by a finite length straightwire can be calculated using Biot-Savart’s Law.

Performing the calculation gives for the magnitude

B =µ0I

4πa(cos θ2 + cos θ1)

Sources of Magnetic Fields

Theory

Biot-Savart

Infinite Length Straight Wire

What is the magnitude of the magnetic field at P if the wire hasinfinite length rather than finite length?

Infinite length will require θ1 → 0 and θ2 → 0.

B = limθ1,θ2→0

µ0I

4πa(cos θ2 + cos θ1)

=µ0I

2πa

Sources of Magnetic Fields

Theory

Biot-Savart

Infinite Length Straight Wire

What is the magnitude of the magnetic field at P if the wire hasinfinite length rather than finite length?

Infinite length will require θ1 → 0 and θ2 → 0.

B = limθ1,θ2→0

µ0I

4πa(cos θ2 + cos θ1)

=µ0I

2πa

Sources of Magnetic Fields

Theory

Biot-Savart

Infinite Length Straight Wire

What is the magnitude of the magnetic field at P if the wire hasinfinite length rather than finite length?

Infinite length will require θ1 → 0 and θ2 → 0.

B = limθ1,θ2→0

µ0I

4πa(cos θ2 + cos θ1)

=µ0I

2πa

Sources of Magnetic Fields

Theory

Biot-Savart

Magnetic Field for Current-Carrying Wire

Infinite Length Wire

The magnitude of the magnetic field a distance a away from a wirewith current I is

B =µ0I

2πa

The magnetic field line direction is given by a right hand rule.

Sources of Magnetic Fields

Questions

Concept Questions

One Current Loop

A current loop lies in the xy-plane and has counterclockwisecurrent. The positive z-axis is coming out of the page. What is thedirection of the magnetic field along the entire z-axis?

1 Into the page

2 Out of the page

3 It switches direction

Correct Answer: 2(Use the right hand rule and remember that magnetic field linescannot cross.)

Sources of Magnetic Fields

Questions

Concept Questions

One Current Loop

A current loop lies in the xy-plane and has counterclockwisecurrent. The positive z-axis is coming out of the page. What is thedirection of the magnetic field along the entire z-axis?

1 Into the page

2 Out of the page

3 It switches direction

Correct Answer: 2(Use the right hand rule and remember that magnetic field linescannot cross.)

Sources of Magnetic Fields

Questions

Concept Questions

Three Current Loops

Three loops with equal radii and equal current magnitude lie onplanes parallel to the yz-plane. What is the direction of themagnetic field at the origin?

1 Positive x direction

2 Negative x direction

3 Zero magnitude (nodirection)

Correct Answer: 1(The effects of the two outer current loops cancel out, leaving themagnetic field generated from the middle loop.)

Sources of Magnetic Fields

Questions

Concept Questions

Three Current Loops

Three loops with equal radii and equal current magnitude lie onplanes parallel to the yz-plane. What is the direction of themagnetic field at the origin?

1 Positive x direction

2 Negative x direction

3 Zero magnitude (nodirection)

Correct Answer: 1(The effects of the two outer current loops cancel out, leaving themagnetic field generated from the middle loop.)