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Electricity and Magnetism Review Notes by L Electricity and Magnetism Review Notes by L. Qian1 Electric Magnetic ============================== ============================= Charge: Current Element: Symbol: q Symbol: sdi r Unit: C [Coulomb] Unit: A⋅m [Ampere-meter] Current: Symbol: i Unit: A (Ampere) --------------------------------------------------- ------------------------------------------------- Electric Field: Magnetic Field: r r Symbol: E Symbol: B Unit: V/m = N/C Unit: T = N⋅s /(C⋅m) [T: Tesla] - Electric field due to a static charge q - Magnetic field due to a current element sdi r r 1 q r μ r × rsid ˆ μ r × rsid r E = rˆ B = 0 = 0 r 2 r 2 r 3 4πε 0 r 4π r 4π r (from Coulomb’s Law) (Biot-Savart Law) y y rˆ rˆ q r rr sid rr P P x x - Magnetic field due to an infinitely long current-carrying straight wire i μ i B = 0 2πR R 1 Notes are for your reference only. Please check with the textbook for accuracy. Notations may be different from that of the textbook. - Electric field in a parallel-plate capacitor: - Magnetic field in an ideal solenoid: V q E == = μ0niB d ε o A (where A is the area of the plate, and (where n is the number of turns per unit d is the plate separation) length) --------------------------------------------------- ------------------------------------------------- Electric Field Lines: Magnetic Field Lines: - Start from positive charges or ∞ - External field lines start from the N-pole - End at negative charges or ∞ - External field lines end at the S-pole - The denser the lines, the higher the - The denser the lines, the higher the E field magnitude B field magnitude - The tangent of the field line at any point - The tangent of the field line at any point indicates the E field direction indicates the B field direction Sample field lines: Sample field lines: note the right-hand rule - Uniform E field inside a parallel-plate - Uniform B field inside an ideal solenoid capacitor (inductor) V+ r E V − = EdV Φ B = BA --------------------------------------------------- ------------------------------------------------------ Magnetic Flux: Symbol: ΦΒ Unit: Wb = T⋅m [Wb: Weber] Definition: r r B ⋅=Φ AdB ∫S Magnetic Flux Linkage (of a solenoid): NΦB (N= number of turns) --------------------------------------------------- ------------------------------------------------------ Electric Force: Magnetic Force: r r Symbol: F Symbol: F Unit: N [Newton] Unit: N [Newton] - Force on a charge in an external E field - Force on a moving charge in an external B field r r r r F= qE F= qvr × B - Force on a current element in an external B field r r r F= idL × B - Force between two charges q1 and q2 - Force between two current-carrying wires ia and ib 1 q q μ ii F = 1 2 F = 0 a b L r 2 ba 4πε 0 r 2π d (note, above formula is the force exerted on wire b over length L) ib ib ia d ia --------------------------------------------------- ------------------------------------------------------ Electric Potential: Symbol: V Unit: V = J/C [V: Volt] - Electric potential due to a single charge q: 1 q V = 4πε 0 r - Potential created by multiple point charges: 1 q V = ∑ i 4πε 0 i ri - Potential difference between capacitor plates: V= q/ C - Equal potential lines are perpendicular to field lines --------------------------------------------------- ------------------------------------------------------ Electric Potential Energy: Symbol: UE Unit: J [Joule] UE = qV or ΔUE = q Δ V --------------------------------------------------- ------------------------------------------------------ Induction (Faraday’s Law): - Induced emf by one turn of coil: dΦ E = − B unit: V dt - Induced emf by a solenoid of N turns: dΦ E = −N B unit: V dt --------------------------------------------------- ------------------------------------------------------ Capacitance: Inductance: Symbol: C Symbol: L Unit: F = C/V [F: Farad] Unit: H = Wb/A [H: Henry] Definition: Definition: q 1 i NΦ C = = or L = B V L NΦ B i - Parallel-plate capacitor - Ideal solenoid of length l ε A C = 0 L= μ n 2 Al d 0 (where A is the area of the plate, and (where n is the number of turns per unit d is the plate separation) length, A is cross-sectional area) - Capacitance is a geometry- and material- - Inductance is a geometry- and material- related parameter related parameter --------------------------------------------------- ------------------------------------------------------ Capacitors in parallel Inductors in parallel 1 1 1 CCCeq =1 + 2 + ... = + +... LLLeq 1 2 (Voltage is the same for all capacitors) (Flux linkage is the same for all inductors) Capacitors in series Inductors in series 1 1 1 = + +... LLLeq = 1+ 2 + ... CCCeq 1 2 (Charge is the same for all capacitors) (Current is the same for all inductors) --------------------------------------------------- ------------------------------------------------------ Energy stored in a capacitor Energy stored in an inductor q2 1 1 U = = CV 2 U= Li 2 E 2C 2 B 2 Electric energy density Magnetic energy density U E 1 2 U B 1 2 uE = = ε o E uB = = B volume 2 volume 2μo ============================== ============================= .
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