LC Energy Transfers In an oscillating LC circuit, energy is shuttled periodically between the electric field of the capacitor and the magnetic field of the inductor; instantaneous values of the two forms of energy are where q is the instantaneous charge on the capacitor and i is the instantaneous current through the inductor. The total energy remains constant. Figure 1: Eight stages in a single cycle of oscillation of a resistanceless LC circuit. (a) Capacitor with maximum charge,no current.(b) Capacitor discharging,current increasing.(c) Capacitor fully discharged,current maximum.(d) Capacitor charging but with polarity opposite that in (a),current decreasing.(e) Capacitor with maximum charge having polarity opposite that in (a),no current.( f ) Capacitor discharging,current increasing with direction opposite that in (b).(g) Capacitor fully discharged,current maximum.(h) Capacitor charging,current decreasing.

LC Charge and Current Oscillations The principle of conservation of energy is given by: Which is the differential equation of LC oscillations (with no resistance). The solution is: in which Q is the charge amplitude (maximum charge on the capacitor) and the angular frequency ω of the oscillations is Figure 2: (a) The potential difference across the capacitor of the circuit of Fig.1 as a function of time. This quantity is proportional to the charge on the capacitor. (b) A potential proportional to the current in the circuit of Fig.1. The letters refer to the correspondingly labeled oscillation stages in Fig.1. Figure 3: The stored magnetic energy and electrical energy in the circuit of Fig. 1 as a function of time. T is the period of oscillation. Note that their sum remains constant.

The phase constant φ is determined by the initial conditions (at t = 0) of the system. The current i in the system at any time t is in which ωQ is the current amplitude I. Oscillations in an LC circuit are damped when a dissipative element R is also present in the circuit. Then The solution of this differential equation is We consider only situations with small R and thus small damping; then Damped Oscillations Figure 4: A series RLC circuit. As the charge contained in the circuit oscillates back and forth through the resistance, electromagnetic energy is dissipated as thermal energy, damping (decreasing the amplitude of) the oscillations.

Alternating Currents; Forced Oscillations A series RLC circuit may be set into forced oscillation at a driving angular frequency by an external alternating emf The current driven in the circuit is where φ is the phase constant of the current. Resonance The current amplitude I in a series RLC circuit driven by a sinusoidal external emf is a maximum when the driving angular frequency equals the natural angular frequency ω of the circuit (that is, at resonance). Then and the current is in phase with the emf. Figure 5: An Alternating Current Generator Figure 6: Single-loop circuit containing a resistor, a capacitor, and an inductor.

The alternating potential difference across a resistor has amplitude the current is in phase with the potential difference. For a capacitor, in which is the capacitive reactance; the current here leads the potential difference by 90° For an inductor, in which is the inductive reactance; the current here lags the potential difference by 90° Single Circuit Elements Figure 7: A resistive load Figure 9: An inductive load Figure 8: A capacitive load

Series RLC Circuits For a series RLC circuit with an alternating external emf given by and a resulting alternating current given by (current amplitude )

Power In a series RLC circuit,the average power of the generator is equal to the production rate of thermal energy in the resistor: rms stands for root-mean-square; the rms quantities are related to the maximum quantities by and The term cos φ is called the power factor of the circuit.

Transformers A transformer (assumed to be ideal) is an iron core on which are wound a primary coil of N p turns and a secondary coil of N s turns. If the primary coil is connected across an alternating-current generator, the primary and secondary voltages are related by The currents through the coils are related by and the equivalent resistance of the secondary circuit, as seen by the generator, is where R is the resistive load in the secondary circuit. The ratio N p / N s is called the transformer’s turns ratio. Figure 10: An ideal transformer

Gauss’ law for magnetic fields, states that the net magnetic flux ( Φ B ) through any (closed) Gaussian surface is zero. It implies that magnetic monopoles do not exist. The simplest magnetic structures are magnetic dipoles. Gauss’ Law for Magnetic Fields Figure 1: A magnet Maxwell’s Extension of Ampere’s Law A changing electric flux induces a magnetic field Maxwell’s law, relates the magnetic field induced along a closed loop to the changing electric flux through the loop. Ampere’s law, gives the magnetic field generated by a current i enc encircled by a closed loop. Maxwell’s law and Ampere’s law can be written as the single equation

Displacement Current The fictitious displacement current due to a changing electric field is defined as The ampere-maxwell law then becomes where i d,enc is the displacement current encircled by the integration loop. The idea of a displacement current allows us to retain the notion of continuity of current through a capacitor. However, displacement current is not a transfer of charge.

Maxwell’s Equations

Earth’s Magnetic Field Earth’s magnetic field can be approximated as being that of a magnetic dipole whose dipole moment makes an angle of 11.5° with Earth’s rotation axis, and with the south pole of the dipole in the Northern Hemisphere. The direction of the local magnetic field at any point on Earth’s surface is given by the field declination (the angle left or right from geographic north) and the field inclination (the angle up or down from the horizontal). Spin Magnetic Dipole Moment An electron has an intrinsic angular momentum called spin angular momentum (or spin) with which an intrinsic spin magnetic dipole moment is associated: Spin cannot itself be measured, but any component can be measured. Assuming that the measurement is along the z axis of a coordinate system, the component can have only the values given by where is the Planck constant. Figure 2: Earth’s magnetic field. The dipole axis MM makes an angle of 11.5° with Earth’s rotational axis RR. Note that the south pole of the dipole is in Earth’s Northern Hemisphere.

Similarly, the electron’s spin magnetic dipole moment cannot itself be measured but its component can be measured. Along a z axis, the component is The energy U associated with the orientation of the spin magnetic dipole moment in an external magnetic field is Orbital Magnetic Dipole Moment An electron in an atom has an additional angular momentum called its orbital angular momentum with which an orbital magnetic dipole moment is associated: Orbital angular momentum is quantized and can have only values given by This means that the associated magnetic dipole moment measured along a z axis is given by The energy U associated with the orientation of the orbital magnetic dipole moment in an external magnetic field is

Diamagnetism Diamagnetism is exhibited by all common materials but is extremely weak that it is masked if the material also exhibits magnetism of either of the other two types (i.e. paramagnetism or ferromagnetism). In diamagnetism, weak magnetic dipole moments are produced in the atoms of the material when the material is placed in an external magnetic field; the combination of all those induced dipole moments gives the material as a whole only a feeble net magnetic field. The dipole moments and thus their net field disappear when is removed. The term diamagnetic material usually refers to materials that exhibit only diamagnetism. Diamagnetic materials develop a magnetic dipole moment directed opposite the external field. Paramagnetism In a paramagnetic material, each atom has a permanent magnetic dipole moment but the dipole moments are randomly oriented and the material as a whole lacks a magnetic field. However, an external magnetic field can partially align the atomic dipole moments to give the material a net magnetic dipole moment in the direction of. If is nonuniform, the material is attracted to regions of greater magnetic field. These properties are called paramagnetism. The alignment of the atomic dipole moments increases with an increase in and decreases with an increase in temperature T. The extent to which a sample of volume V is magnetized is given by its magnetization whose magnitude is Complete alignment of all N atomic magnetic dipoles in a sample, called saturation of the sample, corresponds to the maximum magnetization value For low values of the ratio B ext / T, we have the approximation where C is called the Curie constant.

Ferromagnetism In the absence of an external magnetic field, some of the electrons in a ferromagnetic material have their magnetic dipole moments aligned by means of a quantum physical interaction called exchange coupling, producing regions (domains) within the material with strong magnetic dipole moments. An external field can align the magnetic dipole moments of those regions, producing a strong net magnetic dipole moment for the material as a whole, in the direction of. This net magnetic dipole moment can partially persist when field is removed. If is nonuniform, the ferromagnetic material is attracted to regions of greater magnetic field. These properties are called ferromagnetism. Exchange coupling disappears when a sample’s temperature exceeds its Curie temperature.

Electromagnetic Waves An electromagnetic wave consists of oscillating electric and magnetic fields. The various possible frequencies of electromagnetic waves form a spectrum, a small part of which is visible light. Figure 1: The electromagnetic spectrum

An electromagnetic wave traveling along an x axis has an electric field and a magnetic field with magnitudes that depend on x and t: where E m and B m are the amplitudes of and The electric field induces the magnetic field and vice versa. The speed of any electromagnetic wave in vacuum is c, which can be written as where E and B are the simultaneous magnitudes of the fields. Energy Flow Energy Flow is the rate per unit area at which energy is transported via an electromagnetic wave is given by the Poynting vector

A point source of electromagnetic waves emits the waves isotropically — that is, with equal intensity in all directions. The intensity of the waves at distance r from a point source of power P s is The direction of (and thus of the wave’s travel and the energy transport) is perpendicular to the directions of both The time-averaged rate per unit area at which energy is transported is S avg, which is called the intensity I of the wave: in which Radiation Pressure When a surface intercepts electromagnetic radiation, a force and a pressure are exerted on the surface. If the radiation is totally absorbed by the surface, the force is in which I is the intensity of the radiation and A is the area of the surface perpendicular to the path of the radiation. If the radiation is totally reflected back along its original path, the force is Fig. 2: A point source S emits electromagnetic waves uniformly in all directions.

The radiation pressure p r is the force per unit area: Polarization Electromagnetic waves are polarized if their electric field vectors are all in a single plane, called the plane of oscillation. Light waves from common sources are not polarized; that is, they are unpolarized, or polarized randomly. Polarizing Sheets When a polarizing sheet is placed in the path of light, only electric field components of the light parallel to the sheet’s polarizing direction are transmitted by the sheet; components perpendicular to the polarizing direction are absorbed. The light that emerges from a polarizing sheet is polarized parallel to the polarizing direction of the sheet. If the original light is initially unpolarized, the transmitted intensity I is half the original intensity I 0 :

If the original light is initially polarized, the transmitted intensity depends on the angle θ between the polarization direction of the original light and the polarizing direction of the sheet: Geometrical Optics Geometrical optics is an approximate treatment of light in which light waves are represented as straight- line rays. Reflection and Refraction When a light ray encounters a boundary between two transparent media, a reflected ray and a refracted ray generally appear. Both rays remain in the plane of incidence. The angle of reflection is equal to the angle of incidence, and the angle of refraction is related to the angle of incidence by Snell’s law, where n 1 and n 2 are the indexes of refraction of the media in which the incident and refracted rays travel. Total Internal Reflection A wave encountering a boundary across which the index of refraction decreases will experience total internal reflection if the angle of incidence exceeds a critical angle θ c, where Fig. 3: The angles of incidence θ 1, reflection θ’ 1, and refraction θ 2. Fig. 4: Total internal reflection of light from a point source S in glass occurs for all angles of incidence greater than the critical angle θ c.

Polarization by Reflection A reflected wave will be fully polarized, with its vectors perpendicular to the plane of incidence, if it strikes a boundary at the Brewster angle θ B, where Fig. 5: A ray of unpolarized light in air incident on a glass surface at the Brewster angle θ B.