⚡ Potential Due to a System of Charges (Section 2.5)
Imagine several electric charges sprinkled around. The “height” of electric potential at any point P is just the normal single-charge potential added up for each neighbour, thanks to the superposition principle 😊.
1 • Discrete charges – add ’em all up 🌟
For charges \(q_1,q_2,\dots ,q_n\) sitting at position vectors \(\mathbf r_1,\mathbf r_2,\dots ,\mathbf r_n\) (measured from some origin), the potential at P is
\(V_1=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1}{r_{1P}},\; V_2=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q_2}{r_{2P}},\; \dots\)
and therefore
\(V = V_1+V_2+\dots +V_n =\dfrac{1}{4\pi\varepsilon_0} \left(\dfrac{q_1}{r_{1P}}+ \dfrac{q_2}{r_{2P}}+ \dots + \dfrac{q_n}{r_{nP}}\right).\)
:contentReference[oaicite:0]{index=0}- Only numbers add – the charges’ signs matter, the directions don’t (potential is a scalar).
- Closer charges (small \(r_{iP}\)) contribute more 🚀.
2 • Continuous charge clouds 🌈
When charge is smeared out with density \( \rho(\mathbf r) \), chop space into tiny volumes \(\Delta v\), find each miniature potential, and integrate:
\(V(\mathbf r)=\dfrac{1}{4\pi\varepsilon_0} \displaystyle\int \dfrac{\rho(\mathbf r’)}{|\mathbf r-\mathbf r’|} \,dV’.\)
:contentReference[oaicite:1]{index=1}3 • Special gem – uniformly charged spherical shell 🎯
A shell of total charge \(q\) and radius \(R\) behaves magically:
| Region | Potential \(V\) | Why? |
|---|---|---|
| Outside (\(r\ge R\)) | \(\displaystyle V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}\) | Field is the same as a point charge at the centre. |
| Inside (\(r<R\)) | \(\displaystyle V = \frac{1}{4\pi\varepsilon_0}\frac{q}{R}\) (constant) | Field inside is zero, so moving inside needs no work. |
4 • Worked example – finding zero-potential spots 🍀
Charges \(+3\times10^{-8}\,\text{C}\) and \(-2\times10^{-8}\,\text{C}\) sit 15 cm apart. Setting the far-away potential to zero, the algebra shows two sweet spots where the potentials cancel:
- \(9\;\text{cm}\) from the positive charge (between the charges).
- \(45\;\text{cm}\) from the positive charge on the far side of the negative one.
5 • Quick concept check (Field-line puzzle) 🧐
- Moving a positive test charge against field lines needs outside work; the field’s own work is negative.
- A negative test charge naturally drifts towards higher-potential spots around a positive charge.
- Between two points \(P\) and \(Q\) near a single positive charge, \(V_P>V_Q\) because \(r_P<r_Q\).
- If you push a negative charge from region B to region A (closer to a negative charge), the external work is positive and its kinetic energy drops.
🚀 High-Yield NEET Nuggets
- Superposition rule for potential – just add scalars; no vector headache.
- Potential of a spherical shell – outside acts like a point charge; inside it’s flat-lined (constant).
- Zero-potential points – appear along the line of two unlike charges; solving \(V=0\) is a common exam trick.
- Inside-field-zero ⇒ potential constant – a favourite reasoning step for cavities and shells.
- Sign reasoning with field lines – linking potential differences, work, and kinetic energy directions is a quick-score conceptual question.
👍 Keep practising problems – the math is light once the concepts click!
