Magnetic Forces, Current-Carrying Wires, and Electromagnetic Induction ch 30

Magnetic Forces and Current-Carrying Wires

Particle Identification in Magnetic Fields

Charged particles moving through a magnetic field experience a force that causes them to follow curved paths. This principle is used in devices like bubble chambers to identify particles based on the curvature of their tracks.

• Bubble Chamber: A device filled with superheated liquid where charged particles leave visible tracks of bubbles.

• Magnetic Field Direction: If the field points into the page, positive particles curve counterclockwise, negative particles curve clockwise.

• Radius and Momentum: The radius of curvature r is related to the particle's momentum p by the equation: $r = \frac{mv}{|q|B}$

Example: In a bubble chamber image, the direction and curvature of tracks can be used to determine the sign and momentum of the particles.

Magnetic Force on a Current-Carrying Wire

A wire carrying current in an external magnetic field experiences a force. The magnitude and direction of this force depend on the current, the length of the wire, the strength of the magnetic field, and the angle between the current and the field.

• Magnitude: $F_{\text{wire}} = IlB \sin \alpha$

• Direction: Determined by the right-hand rule and the vector cross product: $\vec{F}_{\text{wire}} = I \vec{l} \times \vec{B}$

Example: For a horizontal wire carrying current to the right in a magnetic field into the page, the force is directed upward.

Forces Between Parallel Wires

Parallel wires carrying currents exert forces on each other due to their magnetic fields. The direction of the force depends on the direction of the currents.

• Same Direction: Wires attract each other.

• Opposite Directions: Wires repel each other.

Example: Two parallel wires with currents in opposite directions will experience a force pushing them apart.

Forces and Torques on Current Loops

Torque on a Current Loop

A current loop in a magnetic field experiences a torque that tends to align the loop's magnetic moment with the field.

• Torque Equation: $\vec{\tau} = \vec{m} \times \vec{B}$

• Magnetic Moment: $\vec{m} = I \vec{A}$, where $I$ is current and $\vec{A}$ is the area vector.

Example: A square loop with current in the counterclockwise direction in a rightward magnetic field will experience forces that cause it to rotate about the y-axis.

Potential Energy of a Magnetic Dipole

The potential energy of a magnetic dipole in a magnetic field depends on the orientation of the dipole relative to the field.

• Potential Energy: $U = -\vec{m} \cdot \vec{B}$

• Lowest energy when $\vec{m}$ is aligned with $\vec{B}$; highest when anti-aligned.

Example: For three orientations of a loop, the one with $\vec{m}$ parallel to $\vec{B}$ has the lowest potential energy.

Applications: DC Motor and Galvanometer

• DC Motor: Converts electrical energy to mechanical energy by reversing current direction to maintain rotation as the magnetic moment aligns with the field.

• Galvanometer: Measures current by the torque exerted on a coil in a magnetic field, causing a needle to deflect over a scale.

Magnetic Properties of Matter

Ferromagnetism

Magnetic properties arise from electron spin and orbital motion. Ferromagnetic materials have atomic dipoles that align over large domains, resulting in strong, persistent magnetization.

• Ferromagnetic Materials: Iron, cobalt, nickel, etc.

• Domains can be aligned by external fields and tend to remain aligned.

Electromagnetic Induction

Induced Current and Motional EMF

An induced current is generated in a conductor when it moves relative to a magnetic field, or when the magnetic field through a loop changes.

• Motional EMF: For a wire of length $l$ moving at speed $v$ perpendicular to a magnetic field $B$: $\mathcal{E} = vlB$

Example: Moving a wire up or down in a magnetic field induces a current; moving left or right does not.

Magnetic Flux

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

• Uniform Field: $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos \theta$

• Non-uniform Field (Surface Integral): $\Phi_B = \int_{\text{surface}} \vec{B} \cdot d\vec{A}$

Example: The flux through a loop depends on the area within the field and the angle between the field and the area vector.

Lenz's Law

Lenz's Law states that the direction of an induced current is such that the induced magnetic field opposes the change in magnetic flux.

• Induced current only occurs if the magnetic flux is changing.

• The induced field always acts to oppose the change (conservation of energy).

Faraday's Law

Faraday's Law quantifies the induced emf due to a changing magnetic flux:

• $\mathcal{E} = \left| \frac{d\Phi_B}{dt} \right|$

Lenz's Law determines the direction of the induced current.

Applications and Examples

• Triple Beam Balance Damping: Eddy currents induced in a metal plate provide damping.

• Magnet Down Tube: A falling magnet induces currents in a conducting tube, slowing its fall.

• Ring and Electromagnet: Induced current in a ring creates a magnetic field that can cause the ring to jump when an electromagnet is switched on.

• Ground Fault Circuit Interrupter (GFCI): Detects imbalance in current due to leakage by sensing changes in magnetic flux.

• Induction Stove: Oscillating currents in a coil induce eddy currents in a pan, heating it directly.

• Metal Detectors and Traffic Light Sensors: Use changing magnetic fields and induced currents to detect metal objects or vehicles.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Magnetic force on wire	$F_{\text{wire}} = IlB \sin \alpha$	Force on wire of length $l$ carrying current $I$ in field $B$
Torque on current loop	$\vec{\tau} = \vec{m} \times \vec{B}$	Torque on loop with magnetic moment $\vec{m}$ in field $\vec{B}$
Motional emf	$\mathcal{E} = vlB$	Emf induced in wire of length $l$ moving at speed $v$ in field $B$
Magnetic flux	$\Phi_B = BA \cos \theta$	Flux through area $A$ at angle $\theta$ to field $B$
Faraday's Law	$\mathcal{E} = \left| \frac{d\Phi_B}{dt} \right|$	Induced emf from changing magnetic flux
Potential energy of dipole	$U = -\vec{m} \cdot \vec{B}$	Energy of dipole $\vec{m}$ in field $\vec{B}$

Additional info: These notes include both conceptual explanations and worked examples, as well as practical applications and demonstrations relevant to Chapters 29 and 30 of a college physics course.