RC Circuits and Magnetic Properties

Magnetic Field

• A moving charge or current creates a magnetic field in the surrounding space.
• The magnetic field exerts a force on any other moving charge or current present in the field.
• A magnet always has a north and a south pole. Experimentally, isolated magnetic poles do not exist; poles always occur in pairs.
• If a bar magnet is broken in two, each broken end becomes a pole.
• Like electric fields, magnetic fields are vector fields, meaning they have both magnitude and direction.
• The symbol B is used to represent magnetic field.
• The direction of a magnetic field is defined as the direction in which a compass needle's north pole would point.
• The SI unit of magnetic field is Tesla (T). Another unit is the Gauss (1 G = 10⁻⁴ T).

Magnetic Force on Moving Charges

• The force on a charge q moving with velocity v in a magnetic field B is given by F = qv × B.
• The SI unit of magnetic field is Tesla (T). Another unit is the Gauss (1 G = 10⁻⁴ T).
• A charge moving parallel to a magnetic field experiences zero magnetic force.
• A charge moving at an angle φ to a magnetic field experiences a magnetic force with magnitude F = |q|v₁B = |q|vB sin φ, where v₁ is the component of v perpendicular to B.
• F is perpendicular to the plane containing v and B.
• A charge moving perpendicular to a magnetic field experiences a maximal magnetic force with magnitude Fmax = qvB.

Magnetic Force Direction

• To find the direction of the magnetic field, draw the velocity (v) and magnetic field (B) vectors with their tails together.
• Use the right hand, point your fingers in the direction of v, curl your fingers towards B. Your thumb points in the direction of the force on a positive charge. For a negative charge, the thumb points in the opposite direction.

Magnetic Flux

• We define magnetic flux through a surface just as we define electric flux.
• The magnitude of the magnetic flux through a surface ФB = ∫ Bcos & dA where ФB is the magnetic flux, B is the magnitude of the magnetic field, dA is the vector element of the surface area and φ is the angle between B and the normal to the surface area.
• The SI unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T⋅m².

Gauss's Law for Magnetism

• Unlike electric charge, you cannot have a single magnetic charge (magnetic monopole).
• The total magnetic flux through any closed surface equals zero.

Magnetic Field Direction of a Current Element

• The magnetic field direction is in a plane perpendicular to the current.
• Use the right-hand rule to determine the direction of the magnetic field. Curl your fingers in the direction of the current, your thumb points in the direction of the magnetic field.

Ampere's Law

• The closed line integral of the magnetic field equals μ₀ times the algebraic sum of the currents enclosed by the path.
• Choose an arbitrary closed curve for the line integral.
• Curl the fingers of your right hand around the integration path in the direction of integration. Your thumb points in the direction of positive current. Currents in the opposite direction are negative.

Field of a Long, Straight, Current-Carrying Conductor

• The magnetic field around a long, straight, current-carrying conductor forms circles.
• Use the right-hand rule to determine the direction of the magnetic field. Point the thumb of your right hand in the direction of the current; your fingers curl around the wire in the direction of the magnetic field lines.

Magnetic Field of a Long Cylindrical Conductor

• A cylindrical conductor with radius R, the current I is uniformly distributed in the cross-sectional area..
• Inside the conductor (r < R): B = μ₀I r/2πR².
• Outside the conductor (r > R): B = μ₀I/2πr

Field of a Solenoid

• A solenoid is a helical winding of wire. The field near the center of a solenoid is nearly uniform over the cross section and parallel to the axis. The external field is very small.
• The magnitude of the field inside is B = μ₀nI, where n is the number of turns per unit length and I is the current.

The Bohr Magneton

• The source of magnetic field inside materials is moving electrons in atoms forming microscopic current loops.
• In some materials, these currents are randomly oriented, thus resulting in zero net magnetic field.
• In other materials, an external field can cause the loops to orient preferentially in the direction of the field, thus magnetizing the material.
• The Bohr Magneton (µB) gives the magnetic strength of a current loop; for an orbiting electron: µ = evr/2m, where e is electron charge, v is orbital speed, and r is orbit radius.
• µB = eh/4πm = 9.274 x 10⁻²⁴ A∙m².

Magnetic Materials - Paramagnetism

• Atoms with unpaired electrons have a net magnetic moment.
• When placed in an external magnetic field, the magnetic moments align with the field, attracting the material to the magnet.
• B = B₀ + μ₀M, where M is the magnetization of the material.
• The relative permeability (Km) of the material is greater than 1.
• Km = µ/µ₀=1 +Xm

Magnetic Materials - Diamagnetism

• Atoms with completely paired electrons have no net magnetic moment.
• When placed in an external magnetic field, an induced magnetic field opposes the applied field, repelling the material from the magnet.
• The relative permeability (Km) of the material is slightly below 1, meaning Xm is negative.

Magnetic Materials - Ferromagnetism

• Strong interactions between atomic magnetic moments cause them to line up parallel within regions called magnetic domains, even without an external field.
• In an external field, the domains tend to align parallel to the field; magnetized domains grow, while others shrink.
• Ferromagnetic materials have a Km much larger than 1.

Alternating Current

• An AC source provides sinusoidally varying voltage or current.
• A sinusoidal voltage is described by v = Vcos(ωt), where V is the voltage amplitude, ω is the angular frequency (equal to 2πf), and t is time.

Resistor in an AC Circuit

• The voltage across a resistor in an AC circuit is in phase with the current.
• VR = IR, where VR is the voltage amplitude across the resistor, I is the current amplitude, and R is resistance.

Inductor in an AC Circuit

• The voltage across an inductor in an AC circuit leads the current by 90 degrees.
• VL = IXL, where VL is the voltage amplitude across the inductor, I is current amplitude, and XL is the inductive reactance (XL = ωL).

Capacitor in an AC Circuit

• The voltage across a capacitor in an AC circuit lags the current by 90 degrees.
• Vc = IXc, where Vc is the voltage amplitude across the capacitor, I is current amplitude, and Xc is the capacitive reactance (Xc = 1/ωC).

R-L Circuits

• In an R-L circuit, the inductor opposes changes in current.
• The current builds up gradually toward a final value of ε/R.
• The time constant is τ = L/R

R-C Circuits

• In an R-C circuit, the capacitor opposes changes in voltage.
• The charge builds up toward a final value of Cε.
• The time constant is τ = RC

L-C Circuits

• An L-C circuit exhibits oscillatory behavior, with charge and current oscillating sinusoidally.
• ω=1/√(LC) is the angular frequency.

Dive into the fundamentals of RC circuits, exploring how capacitors charge and discharge over time. This quiz covers key concepts such as time constants, Kirchhoff's rules, and the magnetic properties of materials, providing a comprehensive understanding of both electrical and magnetic phenomena.