TLDR;
This video provides a list of important derivations for the upcoming ISC 2024 physics exam. It covers key derivations from various chapters, including electrostatics, current electricity, magnetic effects of current and magnetism, electromagnetic induction and alternating current, optics (ray and wave), and clarifies that there are no derivations from the final three chapters.
- Electrostatics: Electric field due to dipole, torque on a dipole, Gauss's theorem applications, electric potential, energy stored in a capacitor.
- Current Electricity: Drift velocity, current density, resistivity, balanced Wheatstone bridge.
- Magnetism: Magnetic field due to current-carrying conductors, Ampere's law applications, force between parallel wires, torque on a current loop, magnetic dipole moment of an electron.
- Electromagnetism: Motional EMF, self and mutual inductance, average and RMS values of current and voltage, impedance in LCR circuits.
- Optics: Prism refractive index, refraction at spherical surfaces, lens maker's formula, microscope and telescope magnification, Huygens' principle, fringe width in interference and diffraction.
Electrostatics [0:22]
The video begins with electrostatics, highlighting the derivation of the electric field for an electric dipole in two positions: on the axis (end-on position) and on the perpendicular bisector (broadside-on position). The formulas for these, E axial = 1/4πε0 * 2p/r³ and E equatorial = 1/4πε0 * p/r³, are to be derived. The next derivation involves torque (τ = p x E) on an electric dipole kept in a uniform electric field (E). The video then moves to applications of Gauss's theorem to obtain the electric field due to an infinitely long straight charged wire (λ / 2πε0r), an infinitely charged plane sheet (σ / 2ε0), and a charged hollow spherical shell (E outside = kQ/r², E surface = kQ/R², E inside = 0). Finally, the electric potential (V) due to a point charge (V = 1/4πε0 * Q/r) and an electric dipole (V axial = k p/r², V equatorial = 0) are discussed, along with the potential energy (U = -p.E) of an electric dipole in a uniform electric field.
Capacitance and Energy Storage [2:50]
The discussion continues with the derivation of the energy stored in a capacitor (½ CV², ½ QV, ½ Q²/C) and energy density (u = ½ ε0E²). The video also covers the capacitance of a parallel plate capacitor when filled completely (C' = kε0A/d) and partially (C' = ε0A / (d - t + t/k)) with a dielectric slab, where 'k' is the dielectric constant, 'A' is the area of the plate, 'd' is the distance between the plates, and 't' is the thickness of the dielectric slab.
Current Electricity [4:06]
Moving on to current electricity, the video emphasises the importance of deriving the expression for drift velocity (vd = eEτ / m). It also includes the derivation of the relationship between current and drift velocity (I = nAvd e). Further derivations include current density (J = nAvd e), resistivity (ρ = m / ne²τ), and conductivity (σ = ne²τ / m). Finally, the derivation for the balanced condition of a Wheatstone bridge (P/Q = R/S) is required, assuming galvanometer current (Ig) is zero.
Magnetic Effects of Current and Magnetism [5:26]
In the chapter on magnetic effects of current and magnetism, the video highlights the application of the Biot-Savart law to find the magnetic field (B) at the centre of a circular current-carrying conductor (B = μ0nI / 2r) and at any point on the axis of a circular current-carrying conductor (B = μ0nIr² / 2(r² + a²)^(3/2)). The use of Ampere's circuital law to derive the magnetic field near an infinitely long straight conductor (B = μ0I / 2πr) and inside a solenoid (B = μ0nI) is also important. The derivation of the force between two long parallel wires carrying current (F/L = μ0I1I2 / 2πr) and the torque on a current-carrying loop placed in a uniform magnetic field (τ = M x B) are also included.
Magnetic Dipole Moment [6:55]
The final derivation in this section is the magnetic dipole moment of an electron in a hydrogen atom (μl = e / 2me * l or μl = n h / 4πme).
Electromagnetic Induction and Alternating Current [7:21]
The chapter on electromagnetic induction and alternating current includes the derivation of motional EMF (e = Blv) and power (P = B²l²v² / R). Derivations for the coefficient of self-inductance of a solenoid (L = μ0n²Al or L = μ0N²A / l) and the coefficient of mutual inductance of two coaxial solenoids (M = μ0n1n2A / l or M = μ0N1N2A / l) are also important. The video then mentions obtaining the mean value of current (I average = 0.637 I0) and voltage (E average = 0.637 E0). The relationship between RMS values of current and voltage (Irms = I0 / √2 and Erms = E0 / √2) must also be derived. Finally, using a phasor diagram for a series LCR circuit, the impedance (Z = √(R² + (XL - XC)²)), peak current (I0 = E0 / Z), and phase angle (tan φ = (XL - XC) / R) need to be derived.
Ray Optics [9:31]
The video transitions to optics, starting with ray optics. The first derivation is that of the prism formula for refractive index (μ = sin((A + Dm) / 2) / sin(A / 2)). Next is refraction at a convex spherical surface (μ2 / v - μ1 / u = (μ2 - μ1) / R). The lens maker's formula (1 / f = (μ - 1) * (1 / R1 - 1 / R2)) is highlighted as very important, from which the lens formula (1 / f = 1 / v - 1 / u) can be derived.
Microscopes and Telescopes [10:50]
The video covers the derivations for microscopes and telescopes. For microscopes, the magnification (m = (v0 / u0) * (1 + (D / fe)) = (-L / fe) * (1 + (D / fe))) when the image is at the least distance of distinct vision (D), and (m = (-L / fe) * (D / fe)) when the image is at infinity. Similarly, for telescopes, the magnification (m = (-fo / fe) * (1 + (fe / D))) when the image is at D, and (m = -fo / fe) when the image is at infinity. Diagrams and derivations for both microscopes and telescopes are crucial.
Wave Optics [11:39]
In wave optics, the video stresses the importance of deriving the laws of reflection and refraction using Huygens' principle. The expression for fringe width (β = λD / d) in the case of interference is also important. Finally, for diffraction, the derivation of the expression for minima positions (a sin θ = nλ) and fringe width (β = λD / a) is required.
Modern Physics [12:32]
The video concludes by stating that there are no derivations required from the final three chapters: dual nature of matter and radiation, atoms and nuclei, and electronic devices. These chapters will focus on graphical representations, numerical applications, and knowledge-based questions. The presenter also mentions that notes for all the derivations discussed in the video are available on the Physics Funnlimited Telegram channel.
