Study Notes for General Physics 2: Magnetic Forces and Magnetic Fields

MAPÚA UNIVERSITY

General Physics 2: PHYO2

Instructor: Ralph Vincent Abalos

Magnetic Forces and Magnetic Fields

Overview

• This section covers magnetic fields, the forces exerted by magnetic fields on moving charges, and the interaction of these forces with electric currents.

Magnetic Fields

• Definition: The needle of a compass is a permanent magnet that has a north magnetic pole (N) at one end and a south magnetic pole (S) at the other.

• The behavior of magnetic poles is analogous to electric charges; like poles repel each other, while unlike poles attract.

• Characteristics of Magnetic Fields:

  • Surrounding every magnet is a magnetic field.

  • The direction of the magnetic field at any point can be determined by the direction indicated by the north pole of a small compass needle placed at that point.

Earth as a Giant Magnet

• Key Features:

  • North magnetic pole vs. North geographic pole

  • Magnetic axis vs. Rotational axis

The Force That a Magnetic Field Exerts on a Charge

Conditions for Magnetic Force

• Requirements: For a charge to experience a magnetic force:

  • The charge must be in motion.

  • The velocity of the charge must have a component that is perpendicular to the direction of the magnetic field.

Right Hand Rule No. 1

• Procedure:

  • Extend the right hand such that the fingers point in the direction of the magnetic field, and the thumb points along the velocity of the charge.

  • The palm faces the direction of the magnetic force acting on a positive charge.

  • If the charge is negative, the direction of the magnetic force is opposite to that indicated by the right hand rule.

Definition of the Magnetic Field

• Magnitude Definition: The magnitude of the magnetic field at any point in space is defined mathematically through the angle θ, where θ is the angle between the velocity of the charge and the magnetic field direction.

• SI Unit of Magnetic Field:

  • Tesla (T) is defined as $1 ext{ T} = 1 rac{ ext{N} imes ext{s}}{ ext{C} imes ext{m}}$

  • Note: $1 ext{ Gauss} = 10^{-4} ext{ T}$

Example: Magnetic Forces on Charged Particles

Problem Description

• A proton in a particle accelerator travels at a speed of $5.0 imes 10^6 ext{ m/s}$ and encounters a magnetic field with a magnitude of $0.40 ext{ T}$ making an angle of $30.0^ ext{°}$ with respect to the velocity of the proton.

  • Questions:

  1. Find the magnitude and direction of the force on the proton.

  2. Find the acceleration of the proton.

  3. Determine the force and acceleration if the particle were an electron.

Given Data

• Velocity: $v = 5.0 imes 10^6 ext{ m/s}$

• Magnetic Field: $B = 0.40 ext{ T}$

• Angle: $heta = 30^ ext{°}$

Solution Steps

1. Magnetic Force Calculation for Proton:

   • Formula: $F = qvB ext{ sin } heta$

   • Substituting in values:
     $F = (1.60 imes 10^{-19} ext{ C})(5.0 imes 10^6 ext{ m/s})(0.40 ext{ T}) ext{ sin } 30^ ext{°}$

   • Calculate the force:

     • $F ext{ (magnitude)} = 1.60 imes 10^{-19} imes 5.0 imes 10^6 imes 0.40 imes 0.5 ext{ N}$

     • This yields a specific force value.

2. Acceleration of Proton:

   • To find acceleration, use $a = rac{F}{m}$ where $m$ is the mass of the proton (approx. $1.67 imes 10^{-27} ext{ kg}$).

   • $a = rac{F}{1.67 imes 10^{-27} ext{ kg}}$ gives the resultant acceleration.

3. Electron Force and Acceleration:

   • The force magnitude remains the same, but the direction reverses since the charge is negative.

   • Use mass of electron (approx. $9.11 imes 10^{-31} ext{ kg}$) to find acceleration similarly.

The Motion of a Charged Particle in a Magnetic Field

Scenarios

1. Velocity Perpendicular to Field Lines

• When the velocity is perpendicular to the magnetic field lines, the motion typically results in circular paths due to magnetic force.

2. Velocity Parallel to Field Lines

• If the velocity is parallel to the field, there is no magnetic force acting on the charged particle; it continues moving linearly without deflection.

3. Charge at Rest

• If the charge is at rest (i.e., velocity $v = 0$), then by definition, there is no resultant force since a magnetic field does not influence stationary charges.

Work Done by Forces

• The electrical force can perform work on a charged particle (causing acceleration), while the magnetic force does not do work on charged particles (it only changes direction).

Conceptual Example: Velocity Selector

• A velocity selector is a measurement device that applies an electric field and a magnetic field to balance the forces acting on the charged particle.

• This balance allows scientists to accurately measure the velocity of the charged particle.

The Force on a Current in a Magnetic Field

Magnetic Force on a Current-Carrying Wire

• The magnetic force exerted on the current in a wire leads to a lateral force; it can push the wire to the right.

Force Formula

• The expression for the force on a current-carrying wire in a magnetic field is given as:

  • $F = I L B ext{ sin } heta$

  • Where:

  • $I$ = Current (A)

  • $L$ = Length of the wire (m)

  • $B$ = Magnetic field strength (T)

  • $heta$ = Angle between the wire and the magnetic field direction

Example: Loudspeaker Mechanics

Problem Description

• Consider a voice coil with:

  • Diameter: $0.0025 ext{ m}$

  • Turns of wire: $55$

  • Magnetic field: $0.10 ext{ T}$

  • Current: $2.0 ext{ A}$

Questions to Determine:

1. Find the magnetic force acting on the coil and the cone.

2. Find the acceleration of the voice coil and cone, assuming a combined mass of $0.0200 ext{ kg}$.

Solution Steps

1. Magnetic Force Calculation:

   • Using $F = I L B ext{ sin } 90^ ext{°}$:

   • $F = (2.0 ext{ A}) (55 imes ext{(circumference of the wire)}) (0.10 ext{ T})$

   • Calculate values resulting in specific force.

2. Acceleration Calculation:

   • Use $a = rac{F}{m}$ with the mass of coil and cone provided.

The Torque on a Current-Carrying Coil

Magnetic Torque Basics

• The effect of magnetic forces on loop coils of wire results in equal magnitude forces but opposite directions acting on different sides of the coil.

• The normal of the loop tends to align with the magnetic field due to the torque effects.

Torsional Calculations

• Net Torque:

  • The overall torque on a current-carrying coil in a magnetic field is determined by:

  • $au = N imes I imes A imes B imes ext{ sin } heta$

  • Where:

    • $N$ = Number of loops

    • $I$ = Current

    • $A$ = Area of the coil $A = l imes w$

    • $B$ = Magnetic field strength

    • $heta$ = Angle between plane of the coil and the magnetic field direction

Example: Analyzing a Current-Carrying Coil

Example Problem

• Given a coil with:

  • Area: $2.0 imes 10^{-4} ext{ m}^2$

  • Turns: $100$

  • Current: $0.045 ext{ A}$

  • Magnetic Field: $0.15 ext{ T}$

Questions:

1. Find the magnetic moment of the coil.

2. Find the maximum torque experienced by the coil.

Solution Steps

1. Magnetic Moment Calculation:

   • $ext{Magnetic Moment} = N imes I imes A$

2. Torque Calculation:

   • Using the torque formula under maximum conditions yields specific values.

Example: Torque on a Rectangular Coil

• Example of a rectangular, 26-turn coil, dimensions $9.6 ext{ cm} imes 5.2 ext{ cm}$ carrying a current of $0.086 ext{ A}$ in a $0.61 ext{ T}$ magnetic field, at an angle of $30^ ext{°}$.

  • Torque formulas applied yield calculated values for torque:

  • $au = N imes I imes A imes B imes ext{ sin } heta$

  • Calculation results are provided in the original example.

Conclusion

• Compiled notes provide insights into electromagnetic force principles, applications in technology (like loudspeakers), and behavior influenced by magnetic fields.

• Comprehensive definitions, formulas, and examples serve as a foundational understanding of magnetic forces and fields in physics.