Công suất mạch điện, cảm ứng và chuyển động hạt mang điện
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| impedance/ɪmˈpiːdəns/ | The complex ratio of voltage to current in an AC circuit element. |
| diode/ˈdaɪəʊd/ | An electronic component that allows current to flow much more readily in one direction than in the other. |
| velocity selector/vəˈlɒsɪti sɪˈlektə/ | A device using crossed electric and magnetic fields to allow particles of a selected speed to pass undeflected. |
Not every resistor sees the battery
- A 24 V battery drives a 2 Ω resistor in series with 3 Ω and 6 Ω in parallel. The parallel pair sees only 12 V — the series resistor takes the rest.
- GRE circuits and magnetism questions test network reduction, AC power, Faraday's law and charged-particle motion.
Reduce, then expand
- Series resistances add; parallel conductances add. $3 \parallel 6 = 2$ Ω, total 4 Ω, current 6 A, branch voltage 12 V, branch currents 4 A and 2 A.
- Check: currents into the node sum (4 + 2 = 6); voltages round each loop sum (12 + 12 = 24).
12 V source, 2 Ω in series with 6 Ω ∥ 3 Ω. Current through the 6 Ω resistor in A?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Total 4 Ω, 3 A, branch voltage 6 V.
AC power
- $I_{\text{rms}} = I_{\max}/\sqrt2$. Average power $P = I_{\text{rms}}^2R = V_{\text{rms}}I_{\text{rms}}\cos\varphi$; ideal $L$ and $C$ dissipate nothing on average.
- Series RLC impedance 阻抗: $|Z| = \sqrt{R^2 + (\omega L - 1/\omega C)^2}$. A single ideal diode 二极管 gives half-wave rectification, not full-wave.
A sinusoidal current with peak 2 A flows through 10 Ω. Average power in W?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$I_{\text{rms}}^2 R = 2 \times 10$.
Faraday's law
- $\Phi = BA\cos\theta$ with $\theta$ from the normal. $\mathcal E = -N\,d\Phi/dt$; current opposes the change in flux, not the field.
- 100 turns, 0.020 m², field rising 0.030 T in 0.10 s: $|\mathcal E| = 100 \times 0.020 \times 0.3 = 0.60$ V.
- Centre of a loop: $B = \mu_0I/(2R)$, times $N$ for a coil.
A 50-turn coil of area 0.04 m² sees a perpendicular field rise by 0.5 T in 0.2 s. Average emf in V?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Faraday's law: $\varepsilon = NA\,\Delta B/\Delta t = 50 \times 0.04 \times 0.5/0.2 = 5$ V.
Charges in magnetic fields
- $|q|vB = mv^2/r$: $r = \dfrac{mv}{|q|B}$, $\omega_c = \dfrac{|q|B}{m}$ — independent of speed. Deuterons and alpha particles both have half the proton's $\omega_c$.
- Velocity selector 速度选择器: undeflected when $qE = qvB$, so $v = E/B$ (forces opposite). $E = 6000$ V/m, $B = 0.20$ T: $v = 3\times10^4$ m/s.
Cyclotron frequency against a proton's
Use ω = |q|B/m to compare particles in one field.
In the same magnetic field, an alpha particle's cyclotron frequency compared with a proton's is
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Twice the charge, four times the mass.
A velocity selector with $E = 3000$ V/m and $B = 0.15$ T passes particles at $2\times10^4$ m/s undeflected.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Particles pass undeflected when $qE = qvB$, that is $v = E/B = 3000/0.15 = 2\times10^4$ m/s.
Recap
- Reduce the network, then find each branch's own voltage and current.
- $P = I_{\text{rms}}^2R$ — do not halve again.
- $\mathcal E = -N\,d\Phi/dt$; $r = mv/|q|B$; $\omega_c = |q|B/m$; selector $v = E/B$.
- impedance: The complex ratio of voltage to current in an AC circuit element.