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: : 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 : - 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 ()

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 = EqF r ×= BvqF - Force on a current element in an external B field r r r ×= BLidF

- Force between two charges q1 and q2 - Force between two current-carrying wires ia and ib 1 qq μ ii F = 21 F = 0 ba 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: = / CqV

- Equal potential lines are perpendicular to field lines

------Electric Potential : Symbol: UE Unit: J []

E = qVU or Δ E Δ= VqU ------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: : 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 = μ nL 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

------ in parallel in parallel 111 eq CCC 21 ++= ... ++= ... eq LLL 21 (Voltage is the same for all capacitors) (Flux linkage is the same for all inductors)

Capacitors in series Inductors in series 111 ++= ... eq = + LLL 21 + ... eq CCC 21 (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 = LiU 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 2 volume 2μo ======