Mistake Master
Student view — seeing the site as a student does
Home Unit 10 · Electric Force, Field, and Potential 10.1·10.2·10.3·10.4·10.5·10.6·10.7 Lesson
Skill Check 0 / 10 complete

Build the hill out of qV, not V

Two tools, one decision. A charge moving between two points converts potential energy into kinetic energy: $\Delta K = -\Delta U = q(V_{\text{start}} - V_{\text{end}})$, with every sign kept. That works everywhere. Constant-acceleration kinematics is the specialist tool, and it works only where the force holds still, which in this unit means the uniform field between parallel plates and nowhere else.

§1

The hill a charge sees is qV, and q can turn it over.

Objects move toward lower potential energy, and potential energy is $U = qV$. For a positive charge that agrees with the language of downhill: it speeds up as it moves toward lower potential. For a negative charge the factor of $q$ flips the landscape over.

Release an electron beside a $100$ V plate with a $0$ V plate across the gap:

$$U = qV = (-e)(100\ \text{V}) < 0 \quad \text{at the high-potential plate},$$

which is the lowest $U$ available to it. So the electron does not race across to the $0$ V plate; it is already at the bottom of its own hill, and moving toward $0$ V would cost it energy. An electron accelerates toward higher potential.

The working rule: find the direction in which $U$ decreases, not the direction in which $V$ decreases, and the motion follows. Those are the same direction only when $q > 0$.

§2

Energy first. It never needs the field to hold still.

The one line that answers most of these problems:

$$\Delta K = q\left(V_{\text{start}} - V_{\text{end}}\right), \qquad \text{so} \qquad \tfrac{1}{2}mv^2 = q\,\Delta V \ \text{from rest}.$$

Nothing in that derivation assumed anything about the path or about how the field varies along it. Get the two potentials however you can, from $kq/r$ near a point charge or from the plate values in a capacitor, subtract, and multiply by the charge with its sign.

A useful unit falls out. A charge $e$ moved through $1$ V gains $1$ electronvolt, $1.6\times10^{-19}$ J, which is why particle energies are quoted in eV and keV.

§3

Kinematics needs a constant force, and near a point charge there is not one.

Compute $a = qE/m$ at the release point beside a point charge and the number is correct, for that instant. Move the charge $3$ cm and $E$ has changed, because $E = kq/r^2$ depends on where you are. So $v^2 = v_0^2 + 2ad$ has no constant $a$ to use and the answer it produces is wrong.

  1. Uniform field, between parallel plates. The force really is constant, so $a = qE/m$ holds for the whole trip and all the constant-acceleration relations apply. Launch a charge sideways and it traces a parabola, exactly as a projectile does.
  2. Anywhere near a point charge or any non-uniform field. Use energy. Get $V$ at the start and at the end, subtract, multiply by $q$.

The tell for the misapplication is a solution that computes one acceleration and then uses it over a distance across which the problem has clearly told you the separation changed.

§4

Worked shape: a charge released near another charge.

A charge $q$ is released from rest at distance $r_1$ from a fixed source $Q$, and you want its speed at $r_2$. The whole solution is four steps and no kinematics.

  1. $V_1 = kQ/r_1$ and $V_2 = kQ/r_2$, each with the sign of $Q$.
  2. $\Delta K = q(V_1 - V_2)$, with the sign of $q$.
  3. Started from rest, so $\tfrac12 mv^2 = \Delta K$.
  4. $v = \sqrt{2\,\Delta K/m}$.

Two checks worth running. If the pair is like-signed and moving apart, $\Delta K$ must come out positive, because they repel. If $\Delta K$ comes out negative for a charge you released from rest, the charge cannot get there at all, and that is a real answer rather than an error: it means the charge turns around first.

§5

Skill Check.

Ten scenarios. Pick the chips that match your answer, then check. A scenario marks complete the first time every part is right. Progress saves on this device.

0 of 10 scenarios complete