AP Physics C: E&M — Unit 4: Statics and Dynamics of Magnetic Fields

Magnetic Field ($\vec{B}$)

Vector fields produced by moving electric charges or intrinsic magnetic moments of elementary particles.

SI Unit for Magnetic Field

Tesla (T). 1 T = 1 \frac{N}{C \cdot m/s} = 10^4 Gauss.

Lorentz Force Formula

( \vec{F}_B = q(\vec{v} \times \vec{B}) ) describes the force on a charged particle moving in a magnetic field.

Right-Hand Rule (RHR) for Forces

A method to determine the direction of the magnetic force on a positive charge using the direction of velocity and magnetic field.

Centripetal Force in Uniform Magnetic Field

The magnetic force acts as the centripetal force causing uniform circular motion of a charged particle.

Cyclotron Radius Formula

( r = \frac{mv}{qB} ) describes the radius of the circular path of a particle in a magnetic field.

Cyclotron Frequency

( \omega = \frac{qB}{m} ) describes how often a charged particle moves in a circular motion in a magnetic field.

Magnetic Forces on Wires

A straight wire carrying current in a magnetic field experiences a force described by ( \vec{F}_B = I(\vec{L} \times \vec{B}) ) where ( I ) is the current.

Torque on a Current Loop

Described by ( \vec{\tau} = \vec{\mu} \times \vec{B} ) where ( \vec{\mu} ) is the magnetic dipole moment.

Biot-Savart Law

Analogous to Coulomb's Law for magnetic fields produced by currents, represented as ( d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \hat{r}}{r^2} ).

Magnetic Dipole Moment

( \vec{\mu} = NIA \hat{n} ) defines the strength and direction of magnetism in a current loop.

Potential Energy of a Dipole

( U = -\vec{\mu} \cdot \vec{B} ) describes the potential energy of a magnetic dipole in a magnetic field.

Long Straight Wire Magnetic Field

The field around an infinite line of current, given by ( B = \frac{\mu_0 I}{2\pi r} ).

Force Between Parallel Wires

Describes how two parallel wires exert forces on each other based on their currents, given by ( \frac{F}{L} = \frac{\mu0 I1 I_2}{2\pi d} ).

Ampère’s Law

Related to magnetic fields and currents, stated as ( \oint \vec{B} \cdot d\vec{l} = \mu0 I{enc} ).

Helical Motion of Charged Particles

When a particle has a component of velocity parallel to the magnetic field, it moves in a helix.

Work Done by Magnetic Forces

Magnetic forces do NO work on charged particles, as they act perpendicular to the motion.

Current in Magnetic Field

A wire carrying current in a magnetic field experiences a force proportional to the current and the field strength.

Uniform Circular Motion

Charged particles experience circular motion in a uniform magnetic field when velocity is perpendicular to the magnetic field.

Torque Principle in Electric Motors

The principle of torque acting on a current loop in a magnetic field, used in electric motors.

Magnetic Field of a Solenoid

For an ideal solenoid, the field inside is ( B = \mu_0 n I ), where ( n ) is the turns per unit length.

Magnetic Field of a Toroid

Describes the magnetic field in a toroid as ( B = \frac{\mu_0 N I}{2\pi r} ).

Direction of Force on Moving Charges

Determined by the right-hand rule, with fingers pointing along velocity and curling towards magnetic field.

Zero Force Condition

The magnetic force is zero if a charged particle is stationary or moving parallel to the magnetic field.

Magnitude of Lorentz Force

Given by ( F_B = |q|vB \sin(\theta) ), where ( \theta ) is the angle between velocity and magnetic field.

Electric vs. Magnetic Forces

Electric forces act on stationary and moving charges, while magnetic forces act only on moving charges.

Centripetal Force Relation

In a magnetic field, ( FB = Fc ) which means the magnetic force provides the required centripetal force for circular motion.

Right-Hand Rule Application

For a loop of current, the right-hand rule can be used to determine the direction of the magnetic dipole moment.

Ampère's Law Applications

Useful for calculating the magnetic fields in symmetric situations, such as long wires or solenoids.

Helical Motion Characteristics

In helical motion, the pitch is constant due to the parallel component of velocity while perpendicular motion causes circularity.

Positive Charge Moving in Magnetic Field

For a positive charge, the direction of force is found using the right-hand rule.

Maxwell's Note on Closed Loops

The net force on a closed current loop in a uniform magnetic field is zero.

Force Equations for Non-Uniform Fields

For a curved wire in a non-uniform magnetic field, the force is calculated by integrating along the path.

Biot-Savart Law Applications

Can be used to find the magnetic field produced by an element of current.

Magnetic Field Strength Natural Units

Permeability of free space, ( \mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot ext{ m/A} ).

Force Due to Parallel Currents

Parallel currents attract each other while anti-parallel currents repel each other.

Role of Magnetic Field Lines

Magnetic field lines emerge from the north pole and enter the south pole, indicating field direction.

Identification of Magnetic Field Sources

Magnetic fields are produced by moving charges and the intrinsic magnetic moments of particles.

Angular Frequency of Charged Particles

Angular frequency ( \omega ) characterizes the motion of charged particles in a magnetic field.

Reference to Zero Work by Magnetic Forces

Magnetic forces do zero work as they are always perpendicular to the direction of motion.

Direction and Magnitude of Magnetic Fields

Determined by factors including the current direction, distance from the wire, and the nature of the current.

Application of Right-Hand Rule for Circular Motion

Used to find direction of magnetic field circles created by current in a wire.

Electric Motor Operation Principle

Electric motors work under the principle of torque exerted on current-carrying loops in a magnetic field.

Magnetic Induction by Point Charges

Biot-Savart law allows for magnetic field calculations resulting from moving electric charges.