Theory — Potential Difference and Energy
1) Potential difference as a line integral
\[
\Delta V = V_B - V_A = -\int_A^B \mathbf{E}\cdot d\mathbf{l}.
\]
This says the potential difference equals the negative work per unit charge done by the electric field along a path from A to B.
2) Conservative (electrostatic) fields → path independence
\[
\nabla\times \mathbf{E} = \mathbf{0}
\quad\Rightarrow\quad
\int_A^B \mathbf{E}\cdot d\mathbf{l}
\text{ depends only on A and B.}
\]
For electrostatics (fixed charges), the field is conservative. That is why you can compute \(\Delta V\) using endpoints only.
3) Point charges: potential and \(\Delta V\)
\[
V(\mathbf{r}) = \sum_i \frac{k\,q_i}{r_i},
\qquad r_i = \|\mathbf{r}-\mathbf{r}_i\|.
\]
\[
\Delta V = V_B - V_A.
\]
If point A is at infinity, you often use \(V(\infty)=0\), so \(\Delta V = V_B\).
4) Uniform fields
\[
\Delta V = -\mathbf{E}\cdot(\mathbf{r}_B-\mathbf{r}_A)
= -(E_x\Delta x + E_y\Delta y).
\]
This is a common model for parallel plates (approximately uniform \(E\) between plates away from edges).
5) Energy
\[
\Delta U = q_{\text{test}}\,\Delta V,
\qquad
W_{\text{field}} = -\Delta U.
\]
If \(\Delta U>0\), you must do positive external work to move the charge against the field.
Note: \(\tfrac12 qV\) appears in energy-storage/assembly contexts (e.g., charging capacitors), not for simply moving a fixed test charge in a fixed external potential.
Worked example (matches “Fill example”)
Source charge \(q=1\times 10^{-6}\,\mathrm{C}\). Point B at \(r=0.1\,\mathrm{m}\). Point A at infinity.
\[
\Delta V = \frac{kq}{r}
\approx \frac{(9\times 10^9)(10^{-6})}{0.1}
= 9\times 10^4\,\mathrm{V}.
\]
For \(q_{\text{test}}=2\times 10^{-6}\,\mathrm{C}\):
\[
\Delta U = q_{\text{test}}\Delta V
= (2\times 10^{-6})(9\times 10^4)
= 1.8\times 10^{-1}\,\mathrm{J}.
\]
