Mapping electric influence
Complex electric systems become manageable by combining two descriptions: the electric field is a vector that predicts force, while electric potential is a scalar that tracks energy.
Learning goals
- superpose electric fields as vectors and potentials as scalars;
- connect field, potential difference, work and potential energy;
- interpret field lines and equipotential surfaces correctly;
- use symmetry and Gauss’s law without overextending it;
- analyse dipoles, conductors, capacitors and numerical field maps.
1. Field and force
The electric field is defined using a small positive test charge: $\mathbf E=\mathbf F/q_0$. A point charge $q$ produces $\mathbf E=kq\hat{\mathbf r}/r^2$, outward for positive $q$ and inward for negative $q$, with $k=1/(4\pi\varepsilon_0)$. A real charge $q_t$ experiences $\mathbf F=q_t\mathbf E$; a negative test charge feels force opposite to $\mathbf E$.
For many sources, $\mathbf E(\mathbf r)=\sum_i\mathbf E_i$. Components must be added with signs or vector geometry. Symmetry can cancel some components even when each individual field is nonzero.
2. Potential and energy
Electric potential is potential energy per charge, $V=U/q_t$. A point charge gives $V=kq/r$, and many sources add algebraically: $V=\sum_i kq_i/r_i$. Potential is easier to superpose because it has no direction, but signs of source charges matter.