🚀 Potential Energy in an External Field

1. Single Charge 🔋

When a charge q sits at position r in an external electric field (with potential V(r)), the stored energy is simply \[ U = q\,V(r). \] Think of it as “work needed to bring q from infinity to that spot.” If you accelerate an electron (charge \(e = 1.6\times10^{-19}\,\text{C}\)) through a potential difference of 1 V, it gains \(e\!\times\!1\text{ V}=1.6\times10^{-19}\,\text{J}\). This energy unit is the electron-volt (1 eV). Handy multiples: 1 keV \(=10^3\) eV, 1 MeV \(=10^6\) eV, 1 GeV \(=10^9\) eV, 1 TeV \(=10^{12}\) eV. :contentReference[oaicite:0]{index=0}

2. Two Charges in an External Field 👯‍♂️

Place charges \(q_1\) at r1 and \(q_2\) at r2. Bringing them in one-by-one needs energy from two sources: the external field and their mutual push/pull. Add everything to get \[ U = q_1\,V(r_1) \;+\; q_2\,V(r_2) \;+\; \frac{1}{4\pi\varepsilon_0}\; \frac{q_1\,q_2}{r_{12}}, \] where \(r_{12}\) is the distance between them. The last term is the familiar pair-energy you already know from Coulomb’s law. :contentReference[oaicite:1]{index=1}

✨ Quick Example

  • Two charges 7 mC and –2 mC 18 cm apart store \(U=-0.7\) J.
  • Separating them to infinity needs +0.7 J of work.
  • If an external field \(E=A/r^2\) with \(A=9\times10^5\text{ N C}^{-1}\text{m}^2\) is present, the total energy becomes –49.3 J (external field alters the tally). :contentReference[oaicite:2]{index=2}

3. Electric Dipole in a Uniform Field 🧭

A dipole has moment \(\mathbf{p}\) (magnitude p) and sits in a uniform field \(\mathbf{E}\). It feels no net force, but a torque \(\boldsymbol{\tau}= \mathbf{p}\times\mathbf{E}\) tries to turn it. Rotating the dipole slowly from angle \(\theta_0\) to \(\theta_1\) stores work, giving potential energy \[ U(\theta) = -\,p\,E\,\cos\theta. \] The zero of energy is chosen at \(\theta = 90^\circ\) to make life easy. Minimum energy happens when the dipole lines up with the field (\(\theta=0^\circ\)), maximum when it points opposite (\(\theta=180^\circ\)). :contentReference[oaicite:3]{index=3}

🔥 Molecular Flip Example

A mole of molecules each with \(p=10^{-29}\,\text{C m}\) in a \(10^6\text{ V m}^{-1}\) field realigns after the field suddenly turns by \(60^\circ\). Energy released as heat: \[ \Delta U = -pE(\cos60^\circ-\cos0^\circ)=3\text{ J}, \] so the sample gives off 3 J. :contentReference[oaicite:4]{index=4}

4. Why “External” Matters 🌎

The field here comes from distant or fixed sources—not from the charges you’re calculating energy for. Because those sources don’t move, you can safely ignore any back-reaction from your little charges. :contentReference[oaicite:5]{index=5}

💡 High-Yield NEET Nuggets

  • \(U = qV\) — link electric potential to energy for any charge.
  • Electron-volt conversions (eV ↔ J); common in quick energy estimates.
  • Total energy for two charges combines external potentials and mutual term \(\tfrac{1}{4\pi\varepsilon_0}\,\tfrac{q_1q_2}{r_{12}}\).
  • Dipole energy \(U=-pE\cos\theta\) and its twin, torque \(\tau=pE\sin\theta\).
  • Work needed to separate/assemble charges shows up often in objective questions.