Electricity and Magnetism

Electricity and magnetism carry about 7 questions (our estimate), second only to mechanics. Seven sub-areas share them: electrostatics and capacitors, DC circuits, thermoelectricity, magnetic force and fields, magnetic materials, induction, and AC. Most numerical questions are a single formula; Section B ones chain two (a cell with internal resistance, an RLC power factor, a charge in a magnetic field).

5.1 Electrostatics

Coulomb's law: F=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}, k=9\times10^9 N m² C⁻²; in a medium divide by \varepsilon_r. Two charges of 1 µC at 1 m: 9\times10^{-3} N. Electric field E=\frac{F}{q}=\frac{kQ}{r^2} (lines start on positive charges, never cross, are perpendicular to conductors' surfaces); between plates E=\frac{V}{d}. Dipole: p=qd; torque pE\sin\theta; energy -pE\cos\theta; axial field \frac{2kp}{r^3}, equatorial \frac{kp}{r^3}.

Gauss's law: \oint\vec E\cdot d\vec A=\frac{q_{enc}}{\varepsilon_0}. Results: infinite line \frac{\lambda}{2\pi\varepsilon_0r}; infinite sheet \frac{\sigma}{2\varepsilon_0}; conductor surface \frac{\sigma}{\varepsilon_0}; charged sphere outside \frac{kQ}{r^2}, inside a conductor 0, inside a uniformly charged solid sphere \frac{kQr}{R^3}.

Potential V=\frac{kQ}{r} (scalar; adds algebraically); E=-\frac{dV}{dr}; work W=q(V_A-V_B); potential energy of two charges \frac{kq_1q_2}{r}; equipotential surfaces are perpendicular to field lines. 1 eV =1.6\times10^{-19} J. Point-charge zero field between like charges 4\muC and 1\muC 30 cm apart is at 20 cm from the larger one.

Capacitors. C=\frac{Q}{V}; parallel plate C=\frac{\varepsilon_0A}{d} (with a dielectric K\frac{\varepsilon_0A}{d}); sphere 4\pi\varepsilon_0R; cylindrical, spherical shells. Series: \frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}, same charge; parallel: C=C_1+C_2, same voltage. Energy U=\frac{1}{2}CV^2=\frac{Q^2}{2C}=\frac{1}{2}QV; energy density \frac{1}{2}\varepsilon_0E^2. Dielectric (inserted with the battery off): C\to KC, V\to\frac{V}{K}, E\to\frac{E}{K}, Q constant, U\to\frac{U}{K}; (battery on: V same, Q\to KQ). Polarisation reduces the field inside the dielectric; displacement D=\varepsilon E. Types: paper, mica, electrolytic (polarised), variable. Sharing charge between capacitors: common V=\frac{C_1V_1+C_2V_2}{C_1+C_2}, with an energy loss.

5.2 DC circuits

Ohm's law V=IR; resistivity R=\rho\frac{l}{A} (\rho rises with T for metals, falls for semiconductors); drift velocity v_d=\frac{I}{nAe}; conductivity \sigma=\frac{1}{\rho}. R_t=R_0(1+\alpha t). Series R=\sum R_i; parallel \frac{1}{R}=\sum\frac{1}{R_i} (6 Ω ∥ 3 Ω = 2 Ω). Power P=VI=I^2R=\frac{V^2}{R}; a 100 W, 220 V bulb has R=484\ \Omega and, on 110 V, gives 25 W. Energy =Pt; H=I^2Rt (Joule's law).

Cell: \varepsilon=V+Ir; current I=\frac{\varepsilon}{R+r}; terminal voltage V=\varepsilon-Ir (cell of 2 V, r=0.5 Ω, R=4.5 Ω: 1.8 V). Maximum power transfer when R=r. Series of n cells n\varepsilon, nr; parallel \varepsilon, \frac{r}{n}.

Kirchhoff: junction rule \sum I=0 (charge conservation); loop rule \sum\varepsilon=\sum IR (energy). Wheatstone bridge balances when \frac{P}{Q}=\frac{R}{S}. Meter bridge, potentiometer (compare emfs, \frac{\varepsilon_1}{\varepsilon_2}=\frac{l_1}{l_2}; measures without drawing current). Galvanometer (moving coil, \theta\propto I): ammeter by a parallel shunt S=\frac{I_gG}{I-I_g}; voltmeter by a series multiplier R=\frac{V}{I}_g-G; an ideal ammeter has zero resistance, an ideal voltmeter infinite. Ohmmeter uses a cell with the galvanometer and a series resistor (zero at right end).

5.3 Thermoelectric effect

Seebeck effect: a loop of two different metals with junctions at different temperatures carries an emf, \varepsilon=aT+\frac{1}{2}bT^2 (a thermocouple). The neutral temperature is where \frac{d\varepsilon}{dT}=0, the inversion temperature is 2T_n-T_c. A thermopile (several couples in series) detects small radiation. Peltier effect: current through a junction absorbs or releases heat (\Pi), the reverse of Seebeck; it makes thermoelectric coolers. Thomson effect: heat is absorbed or released along a single conductor with a temperature gradient. Applications: thermometry of very high and low temperatures, refrigeration.

5.4 Magnetic effect of current

Force on a charge: \vec F=q\vec v\times\vec B (no work done); in a perpendicular field the path is a circle of radius r=\frac{mv}{qB}, period T=\frac{2\pi m}{qB} (independent of speed: the cyclotron); with \vec E also: Lorentz q(\vec E+\vec v\times\vec B). A proton at 10^6 m s⁻¹ in 0.1 T: r\approx10 cm. Force on a conductor: F=BIl\sin\theta. Torque on a coil: \tau=NIAB\sin\theta (moving-coil galvanometer); magnetic moment m=NIA. Hall effect: V_H=\frac{BI}{nqt} gives the sign and density of the carriers and measures B.

Biot–Savart: dB=\frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2}, \mu_0=4\pi\times10^{-7} T m A⁻¹. Long straight wire B=\frac{\mu_0I}{2\pi r} (5 A at 0.1 m: 10^{-5} T); centre of a circular loop \frac{\mu_0NI}{2R}; on the axis \frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}; solenoid B=\mu_0nI (n turns per metre), toroid \frac{\mu_0NI}{2\pi r}. Ampère's law: \oint\vec B\cdot d\vec l=\mu_0I_{enc}. Parallel wires: force per length \frac{\mu_0I_1I_2}{2\pi d}, attractive for currents in the same direction (defines the ampere). 10 A and 20 A at 10 cm: 4\times10^{-4} N m⁻¹.

5.5 Magnetic properties of matter

Earth's magnetism: declination, dip (inclination) \delta, horizontal component B_H; \tan\delta=\frac{B_V}{B_H}; dip is 0 at the magnetic equator and 90^\circ at the poles. Materials: diamagnetic (\chi small negative, weakly repelled: Bi, Cu, water); paramagnetic (\chi small positive, \chi\propto\frac{1}{T} Curie: Al, Pt); ferromagnetic (\chi large positive, domains, lose ferromagnetism above the Curie temperature: Fe, Co, Ni). \mu=\mu_0\mu_r, \mu_r=1+\chi, B=\mu_0(H+M). Hysteresis: the B–H loop; retentivity, coercivity; the loop area is the energy lost per cycle per volume. Soft iron (narrow loop, low retentivity) is for transformer cores and electromagnets; steel (wide loop) for permanent magnets.

5.6 Electromagnetic induction

Faraday: \varepsilon=-N\frac{d\Phi}{dt}, \Phi=BA\cos\theta (weber); Lenz's law (the sign: the induced current opposes the change; conservation of energy). Motional emf \varepsilon=Blv (a rod of 0.5 m at 4 m s⁻¹ in 0.2 T: 0.4 V); rotating rod \frac{1}{2}B\omega l^2. Induced charge \frac{\Delta\Phi}{R}. Eddy currents (braking, induction furnace, laminated cores to reduce them).

AC generator: coil of N turns, area A, rotating at \omega: \varepsilon=NBA\omega\sin\omega t. Self-inductance \varepsilon=-L\frac{dI}{dt}, \Phi=LI; solenoid L=\mu_0n^2Al; energy U=\frac{1}{2}LI^2 (2 H, 0 to 5 A in 0.1 s: emf 100 V, energy 25 J). Mutual inductance \varepsilon_2=-M\frac{dI_1}{dt}. Transformer (ideal): \frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}; step-up raises voltage and lowers current; losses: copper, iron (eddy, hysteresis), flux leakage; efficiency \frac{P_s}{P_p}. High-voltage transmission lowers I^2R loss.

5.7 Alternating currents

I=I_0\sin\omega t; I_{rms}=\frac{I_0}{\sqrt{2}}=0.707I_0 (220 V mains has a peak of 311 V; Nepal's supply is 220 V, 50 Hz); mean value over a half cycle \frac{2I_0}{\pi}. Reactances X_L=\omega L (the voltage leads current by 90^\circ), X_C=\frac{1}{\omega C} (the voltage lags by 90^\circ); resistor in phase. Series LCR: Z=\sqrt{R^2+(X_L-X_C)^2}, \tan\phi=\frac{X_L-X_C}{R}. Resonance at \omega_0=\frac{1}{\sqrt{LC}} (L=1 mH, C=1\,\muF: 5 kHz): Z=R, current maximum, phase zero. Quality factor Q=\frac{\omega_0L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}} (sharpness of resonance). Power P=V_{rms}I_{rms}\cos\phi; power factor \cos\phi=\frac{R}{Z} (R = 30 Ω, X_L=60, X_C=20: Z=50, \cos\phi=0.6); a pure L or C draws no average power (wattless current). Choke coil lowers current with little loss; the LC circuit oscillates with f=\frac{1}{2\pi\sqrt{LC}}.

Formula and fact sheet

Practice questions

Set A — Recall (12 questions)

1. The force between two 1 µC charges 1 m apart in air isa9 Nb1 Nc9\times10^{-3} Nd9\times10^{-6} N
2. Capacitors of 2 µF and 3 µF connected in series have a combined capacitance ofa1.2 µFb5 µFc1.5 µFd6 µF
3. The energy stored in a 10 µF capacitor charged to 100 V isa0.5 Jb5 Jc0.5 mJd0.05 J
4. The equivalent resistance of 6 Ω and 3 Ω in parallel isa9 Ωb2 Ωc3 Ωd4.5 Ω
5. The resistance of a 220 V, 100 W bulb isa484 Ωb2.2 Ωc22000 Ωd220 Ω
6. The magnetic field 0.1 m from a long straight wire carrying 5 A isa10^{-6} Tb10^{-4} Tc10^{-5} Td2\times10^{-6} T
7. A 0.5 m conductor carrying 4 A is perpendicular to a 0.2 T field. The force on it isa0.1 Nb0.8 Nc1.6 Nd0.4 N
8. A coil of 100 turns has its flux changed by 0.01 Wb in 0.1 s. The induced emf isa1 Vb10 Vc100 Vd0.1 V
9. A transformer has 100 primary and 1000 secondary turns; 220 V is applied to the primary. The secondary voltage isa2200 Vb22 Vc22000 Vd220 V
10. The rms value of an AC of peak voltage 311 V is nearlya311 Vb440 Vc220 Vd155 V
11. The resonant frequency of a series LC circuit with L=1 mH and C=1\ \muF is nearlya159 Hzb1 kHzc31.6 kHzd5033 Hz
12. A material with a small positive magnetic susceptibility isadiamagneticbparamagneticcferromagneticdsuperconducting

Set B — Understanding (8 questions)

1. Charges +4\ \muC and +1\ \muC are 30 cm apart. The electric field is zero on the line between them at a distance from the 4\ \muC charge ofa10 cmb15 cmc25 cmd20 cm
2. The potential at a distance of 0.9 m from a 2 µC point charge isa2\times10^4 Vb2\times10^3 Vc2\times10^5 Vd1.8\times10^4 V
3. A parallel-plate capacitor has plate area 100 cm² and separation 1 mm in air (\varepsilon_0=8.85\times10^{-12}). Its capacitance isa8.85 pFb88.5 pFc885 pFd8.85 nF
4. A cell of emf 2 V and internal resistance 0.5 Ω is connected to a 4.5 Ω resistor. The terminal voltage isa2.0 Vb1.6 Vc1.8 Vd0.2 V
5. A galvanometer of 100 Ω and full-scale current 1 mA is to read 1 A. The shunt needed is abouta0.1 Ωb1 Ωc10 Ωd100 Ω
6. A coil of 50 turns, area 0.02 m² carries 2 A in a 0.5 T field. The maximum torque isa0.5 N mb2 N mc10 N md1 N m
7. A 2 H inductor's current rises from 0 to 5 A in 0.1 s. The average induced emf isa10 Vb100 Vc50 Vd200 V
8. In a series circuit R=30\ \Omega, X_L=60\ \Omega, X_C=20\ \Omega. The power factor isa0.8b0.5c0.6d1

Set C — Application (6 questions)

1. A 100 W, 220 V lamp is run on 110 V. Its power isa50 Wb100 Wc12.5 Wd25 W
2. In a balanced Wheatstone bridge P=10\ \Omega, Q=20\ \Omega, R=30\ \Omega. The fourth arm S isa15 Ωb40 Ωc60 Ωd20 Ω
3. Two long parallel wires 10 cm apart carry 10 A and 20 A in the same direction. The force per unit length isa4\times10^{-4} N m⁻¹, attractiveb4\times10^{-4} N m⁻¹, repulsivec2\times10^{-4} N m⁻¹, attractived4\times10^{-5} N m⁻¹, attractive
4. A proton (m=1.67\times10^{-27} kg) moves at 10^6 m s⁻¹ perpendicular to a 0.1 T field. The radius of its path is nearlya1 cmb10 cmc1 md0.1 mm
5. A series RLC circuit has R=10\ \Omega, L=0.1 H and C=100\ \muF. Its quality factor is nearlya31.6b316c3.16d0.316
6. A 0.5 m rod moves at 4 m s⁻¹ perpendicular to a 0.2 T field. The emf across its ends isa0.1 Vb4 Vc0.8 Vd0.4 V