Define electric field and electric potential

WHY do we even need these ideas?

If charge A and charge B interact, we could just write the force between them (Coulomb's law). But that couples the two charges together. Physicists wanted to say: "charge A alone prepares the space; then whatever charge you drop in, you just read off the preparation."

• The field answers: "If I put a +1 C test charge here, how hard and which way is it shoved?"
• The potential answers: "How much work per coulomb did I do to get here from infinity?"

This separation is the source vs. test charge idea, and it is the foundation of all circuit theory (voltage = potential difference).

Electric Field

WHY divide by $q_0$? Because we want a property of the space, not of the visitor. Doubling the test charge doubles the force, so $\vec F/q_0$ stays constant — it describes the location itself.

Deriving the field of a point charge (from scratch)

Start from Coulomb's law — the measured force between two point charges: $\vec{F} = \frac{1}{4\pi\varepsilon_0}\frac{Qq_0}{r^2}\hat{r}$

Now apply the definition $\vec E = \vec F / q_0$ — divide out the test charge: $\boxed{\vec{E} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}}$

Electric Potential

Deriving potential of a point charge (from scratch)

Work is force times distance. To bring $q_0$ from infinity to $r$, we push against the field, so: $W = -\int_\infty^r \vec{F}_{elec}\cdot d\vec{l} = -\int_\infty^r q_0 E\,dr = -\int_\infty^r q_0\frac{kQ}{r^2}\,dr$

Why the minus sign? The applied work equals minus the work done by the electric force (we oppose it).

Integrate ($\int r^{-2}dr = -r^{-1}$): $W = -q_0 kQ\left[-\frac{1}{r}\right]_\infty^r = q_0 kQ\left(\frac{1}{r}-\frac{1}{\infty}\right)=\frac{kQq_0}{r}$

Divide by $q_0$ (definition of $V$): $\boxed{V = \frac{kQ}{r}}$

The link between them

$E = -\frac{dV}{dr} \qquad\Longleftrightarrow\qquad V(r) = -\int_\infty^r E\,dr$

Worked Examples

Common Mistakes

Active Recall

Connections

• Coulomb's Law — the force law we divided by $q_0$ to get both quantities.
• Electric Potential Energy — $U=q_0V$, the energy before we normalise.
• Voltage and Potential Difference — practical use in circuits ($V_i-V_f$).
• Equipotential Surfaces and Field Lines — field lines ⟂ equipotentials.
• Capacitance — built directly from field between plates and potential difference.
• Gauss's Law — a faster route to $\vec E$ for symmetric charge.

Concept Map

Hinglish (regional understanding)