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Halliday Physics Chapter 24

Chapter 24. Electric Potential

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이 문서의 목차
  1. 1. Electric Potential (\(\boldsymbol{U}\)) & Mechanical Energy
  2. 2. Electric Potential (\(\boldsymbol{V}\))
  3. 3. Equipotential Surface
  4. 4. Electric Potential of Point Charge and Dipole
    1. (1) Collection of Point Charges
    2. (2) Potential due to Electric Dipole
    3. (3) Electric Potential Energy of System of Charged Particles
  5. 5. Continuous Charge Distribution
  6. 6. Calculating Electric Field from Electric Potential
  7. 7. Electric Potential of Charged Conductor

1. Electric Potential (\(\boldsymbol{U}\)) & Mechanical Energy

  • 정의: 전기장 내 전하 배치에 따라 계에 저장되는 에너지
  • 일-에너지 정리의 보존력
    • 전기력은 보존력
    • \(\displaystyle \Delta U =-W_{\text{field}} = - \int_{\text{i}}^{\text{f}} F_{\text{e}}\ \mathrm ds\)
    • Electric Potential Difference와 관계: \(\displaystyle \Delta U =-W_{\text{field}} = - \int_{\text{i}}^{\text{f}} F_{\text{e}}\ \mathrm ds\)
    • Mechanical Energy(역학적\(E\) 보존)
      • 외력 X(\(W_{\text{app}} = 0\)): \(\Delta K = - \Delta U = -q \Delta V\)
      • 외력 O: \(\Delta K = -q \Delta V + W_{\text{app}}\)

2. Electric Potential (\(\boldsymbol{V}\))

  • 정의: 단위 전하 당 전기 퍼텐셜 에너지
\[V = -\dfrac{W_{\infty}}{q_0} = \dfrac{U}{q_0}\]
  • Electric Potential Difference(\(\Delta V\))
\[\Delta V = V_{\text{f}} - V_{\text{i}} = \dfrac{\Delta U}{q} = - \int_{\text{i}}^{\text{f}} E \cdot \mathrm ds\]
  • Uniform Electric Field: \(\Delta V = E \Delta s\)
  • 단위
    • Volt: \(1\ \text{V} = 1\ \text{J/C}\)
    • \(1\ \text{N/C} = 1\ \text{V/m}\) (전기장)
    • Electron Volt(\(\text{eV}\)): \(1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}\)

3. Equipotential Surface

  • 정의: 공간상에서 electric potential이 동일한 점들의 집합
  • 특징
    • \(W = -q \Delta V = 0\)
    • Perpendicularity: \(\mathbf{E} \perp \text{Equipotential Surface}\)
    • \(\mathbf{E}\)의 방향: electric potential이 가장 급격히 감소하는 방향

4. Electric Potential of Point Charge and Dipole

  • \(V_{\infty} = 0\)
  • Single Point Charge: \(V = k \dfrac{q}{r} = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{q}{r}\)

(1) Collection of Point Charges

  • Superposition principle: scalar이므로 단순 scalar sum 계산
\[V = \sum_{i} V_{i} = \dfrac{1}{4 \pi \varepsilon_0} \sum_{i} \dfrac{q_i}{r_i}\]

(2) Potential due to Electric Dipole

\[V = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{q_1 q_2}{r}\]

(3) Electric Potential Energy of System of Charged Particles

\[U = W = \dfrac{1}{4 \pi \varepsilon_0} \dfrac{q_1 q_2}{r}\]

5. Continuous Charge Distribution

\[V = \dfrac{1}{4 \pi \varepsilon_0} \int \dfrac{\mathrm dq}{r}\]

6. Calculating Electric Field from Electric Potential

  • 특정 방향 \(s\)로의 \(E_s = - \dfrac{\partial V}{\partial s}\)
  • 직교좌표계
\[E_x = - \dfrac{\partial V}{\partial x} \quad E_y = - \dfrac{\partial V}{\partial y} \quad E_z = - \dfrac{\partial V}{\partial z}\]
  • Gradient 표현
\[E = - \nabla \times V = - \left( \dfrac{\partial V}{\partial x},\ \dfrac{\partial V}{\partial y},\ \dfrac{\partial V}{\partial z} \right)\]
  • Uniform Electric Field에서
\[E = - \dfrac{\Delta V}{\Delta s}\]
  • Spherical Symmetry
\[E_r = - \dfrac{\mathrm dV}{\mathrm dr}\]

7. Electric Potential of Charged Conductor

Electrostatic Equilibrium 상태의 과잉 전하 도체에서

  • Uniform Potential: 내부 및 표면 전체에서 \(V\) 동일
  • Internal Electric Field: \(E = 0\)
  • Surface Electric Field: 수직으로 나옴
  • Surface Charge Distribution: 곡률 반지름 작을수록 \(\sigma\) 증가
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