Siêuposition tĩnh điện, từ thông và vật dẫn
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| superposition/ˌsuːpəpəˈzɪʃn/ | The principle that, in a linear system, the combined effect of independent contributions equals the sum of their separate effects. |
| induced charge/ɪnˈdjuːst tʃɑːdʒ/ | Charge that gathers on a conductor’s surface because of nearby charges. |
| equilibrium/ˌiːkwɪˈlɪbrɪəm/ | A state with no resultant force or resultant moment. |
Zero field, 30 volts
- A charged conducting shell has zero electric field inside its metal, yet the metal sits at 30 V relative to infinity.
- GRE electrostatics tests vector superposition 叠加, flux, and induced charges 感应电荷 on conductors.
Superpose vectors, not magnitudes
- $\mathbf E = \sum \dfrac{Q_i(\mathbf r - \mathbf r_i)}{4\pi\varepsilon_0|\mathbf r - \mathbf r_i|^3}$. Forces of 3 N right and 4 N up combine to 5 N, not 7 N.
- Potential is a scalar sum $V = \sum \dfrac{Q_i}{4\pi\varepsilon_0 r_i}$ — easier, but $V = 0$ does not mean $\mathbf E = 0$.
Adding the magnitudes of perpendicular forces (3 N + 4 N = 7 N) is the classic slip; resolve into components.
Two perpendicular fields of 6 N/C and 8 N/C act at a point. Resultant magnitude in N/C?
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Perpendicular fields add as vectors: $\sqrt{6^2 + 8^2} = 10$ N/C.
Flux through an open plane
- Gauss's law is for closed surfaces. A point charge beside an infinite plane sends half its flux through it: $\Phi = \dfrac{q}{2\varepsilon_0}$, independent of distance.
The flux of a point charge $q$ through an infinite plane near it is
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The plane subtends half the full solid angle.
Conductors and induced charge
- In equilibrium 平衡, $\mathbf E = 0$ inside the metal and the whole conductor is at one potential.
- Charge $Q$ at the centre of a shell (radii $a < b$, net charge $q$): inner surface $-Q$, outer surface $q + Q$, outside field as for $Q + q$ at the centre.
$Q = +2$ nC, $q = -1$ nC, $b = 0.30$ m. Inner surface $-2$ nC, outer $+1$ nC. Metal potential $= \dfrac{9\times10^9 \times 1\times10^{-9}}{0.30} = 30$ V; field in the metal 0.
$Q = +3$ nC at the centre, shell net charge $+1$ nC, outer radius 0.20 m. Potential of the metal in V?
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$9\times10^9 \times 4\times10^{-9}/0.2$.
Charge on the inner surface of that shell, in nC?
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Cancels the enclosed +3 nC.
The potential inside a charged conductor is constant, but need not be zero.
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True. Zero field inside means no potential difference between points, so the potential is the same everywhere inside, here 30 V, not necessarily zero.
Three regions
- Cavity ($r < a$): $E = \dfrac{Q}{4\pi\varepsilon_0r^2}$. Metal: $E = 0$. Outside ($r > b$): $E = \dfrac{Q + q}{4\pi\varepsilon_0r^2}$.
- $V$ is continuous everywhere; $E$ jumps at each charged surface. An off-centre charge still induces $-Q$ inside, but the cavity field is no longer radial.
Which field?
Use the right enclosed charge in each region of a shell.
Recap
- Add field vectors by components; potentials add as scalars.
- Half the flux of a point charge passes through an infinite plane.
- Conductor: $E = 0$ inside, constant (not zero) potential; inner surface carries $-Q$.
- induced charge: Charge that gathers on a conductor’s surface because of nearby charges.