Electric Force, Field, and Potential
Core concepts
Coulomb's law gives the force between point charges: $F=k|q_1q_2|/r^2$ with $k\approx 9\times10^9,\mathrm{N,m^2/C^2}$. Electric field is force per unit positive charge, $E=F/q$, with field lines pointing away from positive and toward negative charge. For a point charge, $E=k|q|/r^2$. Electric potential is a scalar: $V=kq/r$, and potential energy $U=qV$. The work done moving a charge is $W=q\Delta V$; moving along an equipotential requires zero work. For parallel plates, $E=V/d$ (uniform field), and potential decreases in the direction of $\vec E$. Electric field points from high to low potential.
Traps to watch
- Field is a vector (direction matters); potential is a scalar (sign matters, but no direction).
- A region of zero field can still sit at nonzero potential (e.g., inside a charged shell).
- Field and potential fall off differently: $E\propto 1/r^2$ but $V\propto 1/r$ for a point charge.
- For conductors in electrostatic equilibrium, $E=0$ inside and the surface is an equipotential.
- Don't forget to use the test charge sign when computing force direction from $\vec F=q\vec E$.
Worked example
Two charges $+q$ and $-q$ are separated by $d$. At the midpoint, fields from each point the same direction (toward the negative charge), so magnitudes add: $E=2kq/(d/2)^2=8kq/d^2$. Potentials, being scalars, cancel: $V=kq/(d/2)+k(-q)/(d/2)=0$.
Rule of thumb: superpose fields as vectors and potentials as signed scalars.