Pre-Quantum Electrodynamics
Definitionems.es.ep.d
- PM 2.1, 2.2
- Gr 2.3
We have seen that the total work done by moving a test charge (think of a point particle, so its position is well-defined) in the presence of an electrostatic field is independent of the path, and that it is proportional to the test charge. We can thus define a function representing the work per unit charge and called the electrostatic potential
\[ \phi({\bf r}) \equiv -\int_{\cal O}^{\bf r} {\bf E} \cdot d{\bf l} \tag{p}\label{p} \]
where \({\cal O}\) is some agreed-upon reference point (remember that the actual value of the energy isn't really important in physics: we can fix the zero where we want; it's energy differences which matter).
The potential difference between two arbitrary points \({\bf a}\) and \({\bf b}\) is well-defined without the need to specify the reference point,
\[ \phi({\bf b}) - \phi({\bf a}) = - \int_{\bf a}^{\bf b} {\bf E} \cdot d{\bf l} \tag{p_diff}\label{p_diff} \]
Thus, the electrostatic potential is interpreted as the potential energy which a unit charge would have obtained if brought to the specified point from the reference point, in other words the work you need to do on the unit charge to bring it there. The reference point is typically put at infinity. The electrostatic potential coming from a single point charge \(q\) at the origin can then be calculated (taking for convenience \(d{\bf l} = \hat{\bf r} dr\), i.e. moving in purely radially) as
\[ \phi({\bf r}) = -\int_{\infty}^r dr \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2} = \frac{1}{4\pi \varepsilon_0} \frac{q}{r} \tag{p_pc}\label{p_pc} \]
The electrostatic potential moreover inherits the superposition principle from the electric field, so for a distribution of point charges \(q_i\) at positions \({\bf r}_i\), we have
\[ \phi({\bf r}) = \frac{1}{4\pi\varepsilon_0} \sum_{i} \frac{q_i}{|{\bf r} - {\bf r}_i|} \tag{p_pcd}\label{p_pcd} \]
For a continuous charge density in a volume \({\cal V}\), we have
\[ \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal V} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}, \tag{p_vcd}\label{p_vcd} \]
whereas for a surface or line charge distribution, respectively,
\[ \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal S} da' \frac{\sigma({\bf r}')}{|{\bf r} - {\bf r}'|} \tag{p_scd}\label{p_scd} \]
\[ \phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\cal P} dl' \frac{\lambda({\bf r}')}{|{\bf r} - {\bf r}'|} \tag{p_lcd}\label{p_lcd} \]
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Created: 2024-02-27 Tue 10:31