Chapter 26 Direct-Current and Circuits - esistors in Series and Parallel - Kirchhoff s ules - Electric Measuring Instruments - -C Circuits
. esistors in Series and Parallel esistors in Series: V ax I V xy I 2 V yb I 3 V ab V ax + V xy + V yb I ( + 2 + 3 ) V ab /I + 2 + 3 e e + 2 + 3 +... esistors in Parallel: I V ab / I 2 V ab / 2 I 3 V ab / 3 I I + I 2 + I 3 V ab (/ + / 2 + / 3 ) I/V ab / e e + 2 + 3 +...
2. Kirchhoff s ules Junction: point where three or more conductors meet (nodes, branch points). Loop: closed conducting path. Kirchhoff s junction rule: the algebraic sum of the currents into any junction is zero. ΣI
-The junction rule is based on conservation of electric charge. No charge can accumulate at a junction total charge entering the junction per unit time total charge leaving. Kirchhoff s loop rule: the algebraic sum of the potential difference in any loop, Including those associated with emfs and those of resistive elements, must eual zero. ΣV (electrostatic force is conservative) Sign Conventions for Loop ule: emf source (ε) positive (travel from to +) negative (travel from + to -) resistor (I) negative (travel in same direction as I decreasing V) positive (travel in contrary direction to I increasing V) - Travel is the direction that we imagine going around the loop, not necessarily the direction of the current.
3. Electrical Measuring Instruments d Arsonval galvanometer (meter): Coil of wire mounted next to a permanent magnet. Attached to the coil is a spring. Any current passing through the coil will induce magnetism in the coil (magnetic field exerts a torue on the coil ~ current). When the coil turns, spring makes restoring torue ~ angular displacement ~ current. V I fs c I fs current full scale (coil) c resistance of coil
Ammeter: device that measures current, - It can be adapted to measure currents larger than its full scale range by connecting sh (shunt resistor) in parallel (some I bypasses meter coil). I fs current through coil I a I sh + I fs I sh current through sh current measured by ammeter I a V ab I fs c I sh sh (I a -I fs ) sh Voltmeter: device that measures voltage, - It can be adapted to measure voltages larger than its full scale range by connecting s in series with the coil. V v V ab I fs ( c + s )
Ohmeter: device that measures resistance. - The series resistance s is adjusted so that when the terminals x-y are short-circuited ( ), the meter deflects full scale (zero). When nothing is connected between x-y (open circuit, ) there is no current (no deflection). For intermediate values, meter scale is calibrated to read. Potenciometer: device that measures emf of a source without drawing any current from it. - ab connected to terminals of known emf (ε ). A sliding contact (c) is connected through galvanometer (G) to unknown source (ε 2 ). As contact (c) is moved along ab, cb varies proportional to wire length (c-b). To find ε 2 (c) is moved until G shows no deflection (I G ): ε I cb - G calibrated by replacing ε 2 by source of known emf.
4. -C Circuits - Capital letters: V, Q, I (constant) - Lowercase letters: v, i, (varying) Charging a Capacitor: t v bc I vab / / ε t t f I v ab v bc ε Q f /C At an intermediate time, t: ε v ab v ab ε i i + v bc C v bc C ε Q f i ε At t t Q f i f
Charging a Capacitor: i d dt d Cε ε dt ( Cε ) d' ' Cε t dt' (solve by changing variable x Cε) ln Cε Cε t Cε t / Cε e Cε ( t / ) ( t / e Q e ) f i d dt ε e I e t / t /
Time Constant: relaxation time of the circuit time after which the current in the circuit has decreased to /e of I and charge has reached (-/e) of Q f Cε. τ - If small circuit charges uickly. - i never becomes exactly, and never becomes exactly Q f (asymptotic behavior). Discharging a Capacitor: t Q, ε (capacitor discharges through ) i C i d dt d' ' Q t dt' ln Q t
Discharging a Capacitor: Q e t / i d dt Q e e t / t / I - During charging: 2 ε i i + i C Instantaneous rate at which battery delivers energy to circuit i 2 power dissipated in i /C power stored in C Total energy supplied by battery: ε Q f Total energy stored in capacitor: Q f ε/2