(1) $∫ x^n dx = {x^{n+1}}/{n + 1} + c$ (2) $∫ 1/x dx = log|x| + c$ (3) $∫a^x dx = {a^x}/{loga} + c$ (4) $∫e^x dx = e^x + c$ (5) $∫sinx dx = – cosx + c$ (6) $∫cosx dx = sinx + c$ (7) $∫tanx dx = log |secx| + c$ (8) $∫cotx dx = log |sinx| + c$ (9) $∫secx dx = log |secx + tanx| + c$ (10) $∫cosecx dx = log |cosecx – cotx| + c$ (11) $∫sec^2 x dx = tanx + c$ (12) $∫cosec^2 x = – cotx + c$ (13) ...
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(1) $d/{dx} x^n = n x^{n-1}$ (2) $d/{dx}$ k (constant) = 0 (3) $d/{dx}$ sinx = cosx (4) $d/{dx}$ cosx = -sinx (5) $d/{dx} tanx = sec^2 x$ (6) $d/{dx} cotx = – cosec^2 x$ (7) $d/{dx} secx = secx tanx$ (8) $d/{dx} cosecx = – cosecx cotx$ (9)$ d/{dx} a^x = a^x log a $ (10) $d/{dx} logx = 1/x$ (11) $d/{dx} e^x = e^x$ (12) $d/{dx} sin^{-1}x = {1}/{√{1 – x^2}}$ (13) $d/{dx} cos^{-1}x = – {1}/{√{1 – x^2}}$ (14) $d/{dx} tan^{-1}x = 1/{1+x^2}$ (15) $d/{dx} cot^{-1}x = – {1}/{1 + x^2}$ (16) $d/{dx} sec^{-1}x = 1/{x√{x^2 – ...
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sin2θ = 2 sinθ cosθ $cos2θ = cos^2 θ – sin^2 θ$ $cos2θ = 1 – 2 sin^2 θ$ $cos2θ = 2 cos^2 θ – 1$ $tan2θ = {2 tanθ}/{1 – tan^2 θ}$ $sin2θ = {2tan θ}/{1 + tan^2 θ}$ $cos2θ = {1 – tan^2 θ}/{1 + tan^2 θ}$ $sin3θ = 3 sinθ – 4 sin^3 θ$ $cos3θ = 4 cos^3 θ – 3 cosθ$ $tan3θ = {3 tanθ – tan^3 θ}/{1 – 3 tan^2 θ}$ $1 + cos2θ = 2 cos^2 θ$ $1 + sin2θ = (cosθ + sinθ)^2$ $1 – cos2θ = 2 sin^2 θ$ $1 – sin2θ ...
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(1) Gauss’s Theorem: TNEI = Σq (2) No. of tubes of induction: ${q}/{kε_o}$ (3) NEI = ε E TNEI over close surface = ∮εE ds cosθ (4) Sphere: $E = {q}/{4πε} {1}/{r^2}$ Cylinder: $E = {q}/{4πε} {2}/{r}$ (5) Surface charge density, $σ = Q/A$ (6) $E_{outside}$ (any shape) = $σ/{kε_o}$ (7) E (plane sheet) = ${σ}/{2kε_o}$ (8) $f = F/{ds} \text" (Mechanical force) "$ = $1/2 {σ^2}/{kε_o} $ = $1/2 kε_o E^2$ (9) Energy density, $u = {dw}/{dv}$ = $1/2 kε_o E^2$ = ${dU}/{dv}$ (10) $C = Q/v$ or $Q = CV$ (11) parallel plate capacitor, $C = {k A ...
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(1) Kirchoff’s Law: (i) ΣI = 0 (ii) ΣIR + ΣE = 0 (2) Wheatstone’s network: Balanced condition, ${R_1}/{R_2} = {R_3}/{R_4}$ (3) $R = ρ l/A$ (4) Meter Bridge: (i) $X/R = l_x/l_R$ (ii) Kelvin’s Method, ${G}/{R} = l_G/l_R$ (5) Potential Gradient: $V/L = k$ (6) I (r+R) = E (7) $σ = R/L$ (8) E = k l (9) $E_1/E_2 = {l_1 + l_2}/{l_1 – l_2}$ (10) $r = R({l_1 – l_2}/{l_2})$ r -> internal resistance of cell
(1) Ampere’s Law: $∮ \ov B . \ov{dl} = μ_o I$ (2) Straight conductor: ${μ_o}/{4π} {2I}/{r}$ (3) Solenoid: (i) B = $μ_o$ n I (ii) Near end: ${μ_o n I}/{2}$ (4) MCG: $I = ({c}/{nBA}) θ$ c -> spring constant (5) Sensitivity: ${dθ}/{dI} = {nBA}/{c}$ (6) Ammeter: (i) $S = ({I_g}/{I – I_g}) G$ (ii) $S = {G}/{(n-1)}$ $n = {I}/{I_g}$ (iii) $R = ({G}/{G + S}) S$ (7) Voltmeter: $R = V/I_g – G$ R = G(n-1) $n = {V}/{V_g}$ (8) Cyclotron: (i) radius of +ve ion: $r = {mv}/{Bq}$ (ii) Period: $T = 2t ...
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(1) c = λ v (2) Wave number: $1/λ = v/c$ (3) $_1 μ_2 = {μ_1}/{μ_2} = {sin i}/{sin r}$ (4) $_1 μ_2 = {c_1}/{c_2}$ (5) $_1 μ_2 = {λ_1}/{λ_2}$ (6) $μ_g = {v_a}/{v_g} = {λ_a}/{λ_g}$ (7) $μ = tan i_p$ $i_p$ -> polarizing angle (8) ${cos i}/{cos r} = {\text"width of beam in air (medium 1)"}/{\text"width of beam in dense medium (medium 2)"}$ (9) $i_c = sin^{-1} (1/μ)$ $i_c$ -> critical angle (10) Doppler effect: $f = f_o (1 + {v_r}/{c})$ $v_r$ is towards observer -$v_r$ is away from observer
(1) Constructive interference: Phase difference: 2nπ Path difference (∆): nλ (2) Destructive interference: Phase difference: (2m – 1)π Path difference (∆): (2m – 1)λ/2 (3) $X = {λD}/{d}$ X -> fringe width (4) $X_{bright} = {nλD}/{d}$ $X_{dark} = {(2m – 1)λD}/{2d}$ (5) Phase difference: ${2πx}/{λ}$
(1) c = vλ (2) E = hυ = hc/λ (3) (i) $KE_{max} = hυ – ϕ_o $ ϕ -> photoelectric work function (ii) $KE_{max} = e V_o$ $v_o$ -> stopping potential i.e. hυ – ϕ = e $v_o$ Condition: if υ = $υ_o$ then $KE_{max} = 0$ Therefore, $ϕ_o = h υ_o$ (4) Threshold: υ = $υ_o$ & λ < $λ_o$ (5) $1/2 m_e v_{max}_^2 = hυ – ϕ$ = $h(υ – υ_o)$ = $hc (1/λ – 1/λ_o)$ (6) Constants: h = 6.63 x $10^{-34}$ m = 9.1 x $10^{-31}$ c = 3 x $10^8$ m/$s^2$ ...
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(1) Bohr’s Postulate: (i) ${mv^2}/{r} = 1/{4πε_o} {e^2}/{r^2}$ (ii) $mvr = {nh}/{2π}$ (iii) $E = E_h – E_p$ (2) $r = ({h^2 ε_o}/{πm e^2}) n^2$ i.e. $r α n^2$ (3) $v = {e^2}/{2ε_o h n}$ i.e. $v α 1/n$ … speed of electron (4) $E_n = ({m e^4}/{8 ε_o h^2}) 1/n^2$ i.e. $E α 1/n^2$ here, $PE = – {e^2}/{4πε_o h^2}$ = $- {m e^4}/{4πε_o_^2 h^2 n^2}$ $KE = – {e^2}/{8πε_o h^2}$ = $- {m e^4}/{8πε_o_^2 h^2 n^2}$ (5) $1/λ = R [1/p^2 – 1/n^2]$ R = 1.097 x $10^7$ ...
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