Formula Sheet for Physics Electricity and Magnetism

F or m u l ae 5 S h e e t f or Ph y si cs | pg. 8 Electricity and Magnetism 5.3: Capacitors Capacitance: C = q/V 5.1: Electrostatics Coulomb’s law: F~ = 1 q1 q2 4π0 r 2 r̂ q1 1 q 4π0 r 2 r̂ q ~ r) = Electric field: E(~ 1 Electrostatic energy: U = − 4π Electrostatic potential: V = ~ · ~r, dV = − E 0 r q2 d r2 q1 q2 r Spherical capacitor: C = 1 q 4π0 r V (~r) = − Z ~ r ∞ A A ~ E r~ +q −q Parallel plate capacitor: C = 0 A/d ~ · d~r E 4π0 r1 r2 r2 −r1 Cylindrical capacitor: C = −q +q r1 2π0 l ln(r2 /r1 ) r2 l r1 p ~ Electric dipole moment: p~ = qd~ +q −q d Capacitors in parallel: Ceq = C1 + C2 A C1 C2 B Potential of a dipole: V = 1 p cos θ r2 4π0 θ V (r) r Capacitors in series: p ~ Er Field of a dipole: θr Eθ p ~ Er = 2p cos θ 1 , r3 4π0 Eθ = 1 Ceq = 1 C1 + 1 C2 C1 C2 A B Force between plates of a parallel plate capacitor: Q2 F = 2A 0 Energy stored in capacitor: U = 21CV 2 = 1 p sin θ 4π0 r 3 Q2 2C = 1 QV 2 ~ ~τ = ~p × E~ Torque on a dipole placed in E: Energy density in electric field E: U/V = 21 0 E 2 ~ U = −~ Pot. energy of a dipole placed in E: p · E~ Capacitor with dielectric: C = 5.2: Gauss’s Law and its Applications H ~ · dS~ Electric flux: φ = E H ~ · dS~ = qin /0 Gauss’s law: E 5.4: Current electricity Current density: j = i/A = σE Drift speed: vd = EP = q a x 1 eE τ 2 m = i neA Resistance of a wire: R = ρl/A, where ρ = 1/σ Field of a uniformly charged ring on its axis: qx 1 4π0 (a2 +x2 )3/2 0 KA d P ~ E Temp. dependence of resistance: R = R0 (1 + α∆T ) Ohm’s law: V = iR E and V (of a uniformly charged sphere: 1 Qr , for r < R E 4π0 R3 E= 1 Q for r ≥ R 4π0 r 2 , O R (   r2 Q 3 − , for r < R V R2 0R V = 8π 1 Q for r ≥ R 4π0 r , O R E and V of a uniformly charged spherical shell:  0, for r < R E E= 1 Q 4π0 r 2 , for r ≥ R O R ( 1 Q , for r < R R V 4π 0 V = 1 Q 4π0 r , for r ≥ R O Field of a line charge: E = R r r Resistors in parallel: = 1 R1 + 1 R2 Resistors in series: Req = R1 + R2 r Wheatstone bridge: A R1 A R1 R2 σ 0 Electric Power: P = V 2 /R = I 2 R = IV B R2 ↑ G R3 R4 V Balanced if R1 /R2 = R3 /R4 . R2 B R1 σ 20 Field in the vicinity of conducting surface: E = 1 Req r λ 2π0 r Field of an infinite sheet: E = Kirchhoff’s Laws: (i) The Junction Law: The algebraic sum of all the currents directed towards a node is zero i.e., Σnode Ii = 0. (ii)The Loop Law: The algebraic sum of all the potential differences along a closed loop in a circuit is zero i.e., Σloop ∆ Vi = 0. F or m u l ae S h e e t f or Ph y si cs | pg. 9 i Galvanometer as an Ammeter: ig G ~ Energy of a magnetic dipole placed in B: ~ U = −~µ · B i i − ig S ig G = (i − ig )S Hall effect: Vw = R Galvanometer as a Voltmeter: G i ↑ A ig ~ B l Bi ned y w d B VAB = ig (R + G) 5.6: Magnetic Field due to Current C R Charging of capacitors: θ dl~ V Field due to a straight conductor: t d i q(t) R µ0 i B = 4πd (cos θ1 − cos θ2 ) Field due to an infinite straight wire: B = ∆H ∆Q = Peltier heat . charge transferred Force between parallel wires: dF dl = i1 µ0 i 1 i2 2πd i2 d T0 Tn T Ti a Field on the axis of a ring: 1. Thermo-emf: e = aT + 12 bT 2 P i BP = 3. Neutral temp.: Tn = −a/b. 4. Inversion temp.: Ti = −2a/b. Thomson effect: emf e = ∆H ∆Q = Thomson heat charge transferred = σ∆T . µ0 ia2 2(a2 +d2 )3/2 Field at the centre of an arc: B = a µ0 iθ 4πa ~ B Field at the centre of a ring: B = Eit Ampere’s law: where i is current, t is time, Z is electrochemical equivalent, E is chemical equivalent, and F = 96485 C/g is Faraday constant. H µ0 i 2a ~ · d~l = µ0 Iin B Field inside a solenoid: B = µ0 ni, n = Field inside a toroid: B = 5.5: Magnetism N l Charged particle in a uniform magnetic field: l µ0 Ni 2πr r ~ = q~v × B ~ + qE ~ Lorentz force on a moving charge: F T = i θ a Faraday’s law of electrolysis: The mass deposited is 1 F ~ B d 2. Thermoelectric power: de/dt = a + bT . mv , qB µ0 i 2πd e Seeback effect: r= ~ ⊗B θ1 Time constant in RC circuit: τ = RC v q ~ r θ2 C Discharging of capacitors: q(t) = q0 e− RC Peltier effect: emf e = ~ ⊗B i µ0 i d~l×~ r 4π r 3 ~= Biot-Savart law: dB h i t q(t) = CV 1 − e− RC m = Zit = x z ~2 B Field of a bar magnet: S 2πm qB d N ~1 B d ~ ⊗r B B1 = ~ B Force on a current carrying wire: B2 = µ0 M 4π d3 Angle of dip: Bh = B cos δ ~ l ~ F µ0 2M , 4π d3 Horizontal i ~ F~ = i ~l × B Magnetic moment of a current loop (dipole): µ ~ A ~ ~ ~µ = iA i ~ ~τ = ~µ ×B~ Torque on a magnetic dipole placed in B: δ Bv Tangent galvanometer: Bh tan θ = µ0 ni , 2r Moving coil galvanometer: niAB = kθ, Time period of magnetometer: T = 2π ~ = µH ~ Permeability: B Bh B i = K tan θ k θ i = nAB q I MBh F or m u l ae S h e e t f or Ph y si cs | pg. 10 C 5.7: Electromagnetic Induction H ~ · dS~ Magnetic flux: φ = B Faraday’s law: e = RC circuit: Lenz’s Law: Induced current create a B-field that opposes the change in magnetic flux. Z= Motional emf: e = Blv ~ v l Z 1 ωC ˜ tan φ = φ R √ R + ω 2 L2 , ˜ tan φ = i di −L dt Z C ωL R R q R2 + νresonance =  1 − ωC q 2 ωL , 1 ωC Z φ ˜ e0 sin ωt i R φ ωL e0 sin ωt R2 L Z= 1 ωCR L i LCR Circuit: Self inductance of a solenoid: L = µ0 n2 (πr 2 l) i h t Growth of current in LR circuit: i = Re 1 − e− L/R R R2 + (1/ωC)2 , ⊗B ~ − L p LR circuit: + e= R e0 sin ωt Z= dφ − dt Self inductance: φ = Li, i ωL tan φ = 1 ωC − ωL N2 ˜ R 1 ωC −ωL R 1 LC 1 2π Power factor: P = erms irms cos φ e 0.63 R e i S Transformer: t L R t Decay of current in LR circuit: i = i0 e− L/R L i0 0.37i0 S i t L R Time constant of LR circuit: τ = L/R Energy stored in an inductor: U = 21 Li2 U V Energy density of B field: u = = B2 2µ0 di e = −M dt Mutual inductance: φ = Mi, EMF induced in a rotating coil: e = NABω sin ωt i Alternating current: t T T = 2π/ω RT Average current in AC: ī = T1 0 i dt = 0 i = i0 sin(ωt + φ), RMS current: irms = h R 1 T T 0 i2 dt Energy: E = irms 2 RT Capacitive reactance: Xc = 1 ωC Inductive reactance: XL = ωL Imepedance: Z = e0 /i0 i1/2 = i0 √ 2 = e1 , e2 e1 i1 = e2 i2 e1 ˜ N1 i1 i2 √ Speed of the EM waves in vacuum: c = 1/ µ0 0 i R N1 N2 i2 t T e2