Chapter 1 / 1.7 Electric Field

Electric Field 🌟

Imagine placing a small positive “test” charge \(q\) at some point P around a fixed charge \(Q\). What you feel there is not mysterious action-at-a-distance; it is the electric field \( \mathbf E \) already present in space. For a point charge at the origin, the field is \[ \mathbf E(\mathbf r)=\frac{1}{4\pi\varepsilon_0}\,\frac{Q}{r^{2}}\;\hat{\mathbf r} \tag{1.6} \] where \( \hat{\mathbf r} \) points radially from \(Q\) toward P. :contentReference[oaicite:0]{index=0}

If you now drop the test charge into that location, the field pushes with a force \[ \mathbf F(\mathbf r)=q\,\mathbf E(\mathbf r)\tag{1.8} \] so \( \mathbf E = \mathbf F/q \). This gives the SI unit of electric field: newton per coulomb (N C^-1). :contentReference[oaicite:1]{index=1}

Key Features 🤔

• Source vs. Test Charge – \(Q\) (source) makes the field; a very small \(q\) (test) merely “samples” it so \(Q\) stays put. The limit \(\displaystyle \mathbf E=\lim_{q\to0} \mathbf F/q\) guarantees \( \mathbf E \) is independent of the tester. :contentReference[oaicite:2]{index=2}
• Direction – Field arrows point outward from a positive source and inward toward a negative one. :contentReference[oaicite:3]{index=3}
• Spherical symmetry – At every point on a sphere of radius \(r\) around a point charge, \(|\mathbf E|\) is the same. :contentReference[oaicite:4]{index=4}
• Superposition – For many charges \(q_i\) at positions \(\mathbf r_i\), fields just add: \[ \mathbf E(\mathbf r)=\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{n} \frac{q_i}{r_{iP}^{\,2}}\;\hat{\mathbf r}_{iP} \tag{1.10} \] where \(r_{iP}\) is the distance from \(q_i\) to point P. :contentReference[oaicite:5]{index=5}
• Why Fields Matter – When charges move, information travels at the finite speed \(c\). Thinking in terms of a field—rather than instantaneous forces—naturally explains the time delay and even allows the field itself to carry energy away as electromagnetic waves. ⚡ :contentReference[oaicite:6]{index=6}

Worked Examples 🧮

Example 1.7 – Falling Electron vs. Proton
A uniform field \(E = 2.0\times10^{4}\,\text{N C}^{-1}\) acts over \(h = 1.5\text{ cm}\).

• Electron acceleration: \(a_e = eE/m_e\); time of fall \(\displaystyle t_e=\sqrt{2h/a_e}\approx 2.9\times10^{-9}\,\text{s}\).
• Proton acceleration: \(a_p = eE/m_p\); time of fall \(t_p\approx 1.3\times10^{-7}\,\text{s}\).
• Take-away: Unlike free-fall in gravity, electric acceleration depends on mass, so heavier particles fall slower in the same field. :contentReference[oaicite:7]{index=7}

Example 1.8 – Fields of Two Opposite Charges
Two charges \(+10^{-8}\,\text{C}\) and \(-10^{-8}\,\text{C}\) are \(0.1\text{ m}\) apart.

• At mid-point A: fields add → \(7.2\times10^{4}\,\text{N C}^{-1}\) to the right.
• At point B (closer to +Q): net \(3.2\times10^{4}\,\text{N C}^{-1}\) to the left.
• At point C (above the line): resultant \(9\times10^{3}\,\text{N C}^{-1}\) to the right (vector addition of two equal-magnitude slanted fields). :contentReference[oaicite:8]{index=8}

Quick-Look Properties 📌

High-Yield Ideas for NEET 🚀

1. The inverse-square form of \( \mathbf E \) for a point charge and its radial directions.
2. Relation \( \mathbf F = q\mathbf E \) and how the unit positive “test” charge frames the concept.
3. Superposition principle for evaluating fields of multiple charges quickly.
4. Distinction between source and test charges, ensuring \( \mathbf E \) stays independent of the test charge.
5. Practical understanding that electric acceleration depends on charge-to-mass ratio, unlike gravitational free-fall (Example 1.7).

Keep these nuggets handy, practice vector addition of fields, and you’ll zap through NEET problems with confidence! ⚡😊