Magnetic Field and Magnetic Forces – Chapter 27 Study Notes

Magnetic Field and Magnetic Forces

Introduction to Magnetism

Magnetism is a fundamental physical phenomenon arising from the motion of electric charges. The most familiar examples are permanent magnets, which can attract or repel other magnets and unmagnetized iron objects. The behavior of a compass needle aligning with Earth's magnetic field is a classic demonstration of magnetism in action.

• Permanent magnets have persistent magnetic properties and can interact with other magnets or magnetic materials.

• Magnetism fundamentally arises from the interaction of moving electric charges.

• Magnetic forces act only on moving charges, distinguishing them from electric forces.

• Example: A compass needle aligns with Earth's magnetic field due to the movement of charges within the needle.

Magnetic Poles

Magnets possess two distinct poles: north (N) and south (S). These poles are responsible for the attractive and repulsive interactions between magnets.

• The north pole of a magnet points toward Earth's geographic north.

• The south pole points toward Earth's geographic south.

• Opposite poles attract; like poles repel.

• Example: Two bar magnets placed with opposite poles facing each other will attract, while like poles will repel.

Magnetism and Certain Metals

Some metals, such as iron, nickel, and cobalt, are attracted to magnets even if they are not themselves magnetized. This property is called ferromagnetism.

• Unmagnetized iron objects are attracted by either pole of a magnet.

• Example: The steel door of a refrigerator is attracted to a magnet regardless of the pole facing it.

Magnetic Field of the Earth

Earth itself acts as a giant magnet, with its magnetic field influencing compass needles and navigation.

• Earth's north geographic pole is near its magnetic south pole.

• The magnetic axis is not perfectly aligned with the geographic axis, causing magnetic declination (variation in compass readings).

• The angle of the magnetic field relative to the surface is called magnetic inclination.

• Example: Compass readings deviate from true north due to magnetic declination.

Magnetic Monopoles

Unlike electric charges, magnetic poles always come in pairs. There is no experimental evidence for the existence of magnetic monopoles.

• Breaking a magnet results in two smaller magnets, each with both a north and south pole.

• Additional info: The search for magnetic monopoles is an ongoing area of research in theoretical physics.

Electric Current and Magnets

Electric currents produce magnetic fields, which can interact with nearby magnets or compass needles.

• A compass near a wire with no current points north.

• When current flows through the wire, the compass needle deflects, indicating the presence of a magnetic field.

• Example: The deflection of a compass needle near a current-carrying wire demonstrates the magnetic effect of electric current.

The Magnetic Field

A magnetic field is a vector field created by moving charges (currents) and exerts forces on other moving charges.

• Symbol: B is used to denote magnetic field.

• The direction of B at any point is the direction a compass needle's north pole points.

• Magnetic fields are vector quantities, having both magnitude and direction.

Characteristics of Magnetic Force on a Moving Charge

The magnetic force on a moving charge has several key properties:

• Proportional to the charge magnitude: Doubling the charge doubles the force.

• Proportional to the magnetic field strength: Doubling the field doubles the force.

• Acts only on moving charges: Stationary charges experience no magnetic force.

• Direction: Always perpendicular to both the velocity (v) and the magnetic field (B).

• Zero force when velocity is parallel or antiparallel to the field.

The Magnetic Force on a Moving Charge

The magnitude of the magnetic force depends on the component of velocity perpendicular to the magnetic field.

• If the particle is at rest or moving parallel to the field, the force is zero.

• Equation: $F = |q| v B \sin \phi$ where $\phi$ is the angle between v and B.

• The force is perpendicular to the plane containing v and B.

Magnetic Force as a Vector Product

The magnetic force is best described using the vector (cross) product:

• Equation: $\vec{F} = q \vec{v} \times \vec{B}$

• This representation captures both the magnitude and direction of the force.

Right-Hand Rule for Magnetic Force

The right-hand rule is a mnemonic for determining the direction of the magnetic force on a positive charge.

1. Place the velocity and magnetic field vectors tail to tail.

2. Rotate v toward B through the smaller angle.

3. Curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the force.

• If the charge is negative, the force direction is opposite to that given by the right-hand rule.

Equal Velocities But Opposite Signs

Charges of equal magnitude but opposite sign moving with the same velocity in a magnetic field experience forces of equal magnitude but opposite direction.

• Equation: $\vec{F}_+ = q \vec{v} \times \vec{B}$, $\vec{F}_- = -q \vec{v} \times \vec{B}$

Unit of Magnetic Field (Tesla)

The SI unit of magnetic field is the tesla (T).

• Equation: $1\ \text{tesla} = 1\ \text{T} = 1\ \frac{\text{N}}{\text{A} \cdot \text{m}}$

• Named after Nikola Tesla.

• Another unit: gauss (G), where $1\ \text{G} = 10^{-4}\ \text{T}$.

Cathode-Ray Tube (CRT)

CRT devices use magnetic fields to deflect electron beams, creating images on screens.

• Electron beams are deflected by magnetic forces.

• Application: Used in older television and oscilloscope screens.

Force on a Moving Charge Due to Electric and Magnetic Fields

When both electric and magnetic fields are present, the total force on a moving charge is the sum of the electric and magnetic forces.

• Equation: $\vec{F} = q \vec{E} + q \vec{v} \times \vec{B}$

• This is known as the Lorentz force.

Magnetic Field Lines

Magnetic fields can be visualized using magnetic field lines, which indicate the direction and strength of the field.

• Field lines are tangent to the direction of B at every point.

• Field lines never intersect.

• Field lines are not lines of force; the force on a charged particle is not along the direction of a field line.

Magnetic Field of a Straight Current-Carrying Wire

A current-carrying wire produces a magnetic field that circles the wire. The direction of the field can be represented using dots (out of the plane) and crosses (into the plane).

• Right-hand rule: Thumb in direction of current, fingers curl in direction of magnetic field.

• Example: Magnetic field around a wire in the plane of the paper.

Magnetic Field Lines of Two Permanent Magnets

Iron filings align along magnetic field lines, visually demonstrating the field pattern between two magnets.

• Field lines emerge from the north pole and enter the south pole.

• Field lines are denser where the field is stronger.

Magnetic Flux

Magnetic flux quantifies the amount of magnetic field passing through a given area.

• For an area element $dA$, the magnetic flux is $d\Phi_B = \vec{B} \cdot d\vec{A}$.

• Total flux through a surface: $\Phi_B = \int \vec{B} \cdot d\vec{A}$

• For any closed surface, the net magnetic flux is zero: $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss's law for magnetism).

Units of Magnetic Field and Magnetic Flux

Quantity	SI Unit	Symbol
Magnetic Field	tesla	T
Magnetic Flux	weber	Wb
Gauss (non-SI)	$1\ \text{G} = 10^{-4}\ \text{T}$	G

• Earth's magnetic field: approximately $10^{-4}\ \text{T}$ or $1\ \text{G}$.

• $1\ \text{Wb} = 1\ \text{T} \cdot \text{m}^2$

Motion of Charged Particles in a Magnetic Field

Charged particles moving in a magnetic field experience a force perpendicular to their velocity, resulting in circular or helical motion.

• The force does not change the speed, only the direction.

• Equation for circular motion: $F = qvB = m \frac{v^2}{r}$

• Radius of the path: $r = \frac{mv}{qB}$

• Example: Electrons in a cathode-ray tube follow curved paths due to magnetic fields.