Power, Capacitors, Magnetism & Photoelectric Effect

Power and Photon Emission

• Power is defined as total energy divided by time, where this energy is carried by photons.
• The energy of a single photon is given by hν, where h is Planck's constant and ν is the frequency.
• The total energy is the product of the number of photons (n) and the energy of each photon (hν).
• Power can be calculated using the formula: P = (n * hν) / t.
• The number of photons emitted per second (n/t) can be found using: P / hν.
• Frequency (ν) is related to the speed of light (c) and wavelength (λ) by the equation: ν = c / λ.

Capacitor Behavior

• A fully charged capacitor opposes current flow.
• Conduction current in a fully charged capacitor is zero.
• Displacement current in a fully charged capacitor is zero.
• The absence of current flow is because the electric flux is not changing in a fully charged capacitor.

Magnetic Properties

• Diamagnetism is a universal property of all substances.
• Every object exhibits diamagnetism.

Photoelectric Effect

• Photon energy is used for the work function (Φ) and imparting maximum kinetic energy (KEmax) to emitted electrons.
• The maximum kinetic energy is given by: KEmax = hν - Φ, and also by KEmax = eV0, where V0 is the stopping potential and e is the electron charge.
• The ratio of maximum kinetic energies for two different materials A and B is: KEa/KEb = (hν - Φa) / (hν - Φb).

Charge Distribution in Hollow Sphere

• Consider a hollow conducting sphere with inner radius r and outer radius 2r.
• A point charge q placed at the center induces a charge of -q on the inner surface.
• A charge of +q resides on the outer surface of the sphere.
• The inner surface charge density is: σinner = -q / (4πr²).
• The outer surface charge density is: σouter = +q / (16πr²).
• The ratio of the inner to outer surface charge densities is -4:1.

Wave Reflection

• When a wave reflects from a rarer to a denser medium, it undergoes a phase change of 180 degrees.
• A crest reflects as a trough due to the phase change.

Electric Field Lines

• Electric field lines originate from positive charges and terminate on negative charges.
• The density of electric field lines indicates the magnitude of the charge; higher density means greater magnitude.

de Broglie Wavelength

• The de Broglie wavelength is given by: λ = h / p, where p is momentum.
• It can also be expressed as: λ = h / √(2m * KE).
• de Broglie wavelength can be calculated using voltage: λ = h / √(2m * qV), where V is voltage, q is charge, and m is mass.
• de Broglie wavelength is inversely proportional to the square root of mass.
• The slope in a λ vs 1/√V graph is inversely proportional to √(2m).
• A greater mass corresponds to a smaller slope in the λ vs 1/√V graph.

X-rays and de Broglie Wavelength Derivation

• Maximum kinetic energy (KEmax ) is approximately equal to hν when the work function is negligible; KEmax = hc / λ.
• This KE value can be used in the de Broglie wavelength formula: λ = h / √(2m * KE).

LCR Circuits

• Impedance in an LCR circuit is: Z = √(R² + (XL - XC)²).
• Inductive reactance is: XL = ωL, where ω is angular frequency and L is inductance.
• Capacitive reactance is: XC = 1 / ωC, where C is capacitance.
• Peak current is: I0 = V0 / Z, where V0 is the peak EMF and Z is impedance.
• The RMS value of voltage is: Vrms = V0/√2.
• Maximum potential difference across a resistor is: Vmax = I0 * R.
• Voltage across the entire LCR series combination is: V = √(VR² + (VL - VC)²).

Capacitor in DC Circuit

• A pure capacitor has a capacitance measured in Farads.
• Frequency in a DC circuit is 0.
• Reactance/resistance in a DC circuit is infinite.

Wavelength and Frequency

• The relationship between frequency and wavelength is: λ = c/f = cT, where T is the period and f is the frequency.

Total Internal Reflection and Critical Angle

• The critical angle (θc) is defined by: sin θc = 1/n, where n is the refractive index.
• Alternatively, sin θc = n2/n1, where n2 < n1.
• Total internal reflection (TIR) occurs when the angle of incidence is greater than θc.
• Refraction occurs when the angle of incidence is less than θc.

Stopping Potential Graph

• The slope of the Stopping Potential (V0) vs 1/λ graph is: Slope = hc/e.
• Graph formulas can be represented as: y = mx + c (linear math formula for graphs to visualize).

Assertion and Reason: Insulators

• Assertion: Insulators have a very large forbidden gap. (TRUE)
• Reason: Valence electrons in insulators are tightly bound. (TRUE)
• Tight binding of electrons prevents them from being free, thus making the material an insulator.

Copper and Magnetic Fields

• Assertion: Magnetic field lines concentrate inside a copper bar (FALSE).
• Reason: Copper is paramagnetic (FALSE).
• Diamagnetism causes magnetic field lines to be repelled from copper.