Describe simple electric field patterns, including the direction of the field: (a) around a point charge (b) around a charged conducting sphere (c) between two oppositely charged parallel conducting plates (end effects will not be examined)
4.2.1 Electric Charge & Electric‑Field Patterns
Learning Objective
Students will be able to describe the electric‑field patterns and state the direction of the field for:
- a point charge,
- a charged conducting sphere,
- two oppositely charged parallel conducting plates (edge effects ignored).
1 Fundamental Concepts
1.1 Electric charge
- Definition: a property of matter that can be transferred between objects.
- Unit: the coulomb (C).
1 C = 6.24 × 1018 elementary charges (e).
- Elementary charge: \(e = 1.60 × 10^{-19}\) C. Charge is always an integer multiple of e (quantisation).
- Conservation of charge: the total charge of an isolated system never changes.
1.2 Production of charge by friction (AO2)
When two different materials are rubbed together, electrons are transferred from one surface to the other – protons remain in the nuclei.
- Material that gains electrons becomes negatively charged.
- Material that loses electrons becomes positively charged.
Simple experiment – Rub a glass rod with silk (or a plastic rod with wool). The rod becomes charged; bring it near small pieces of paper. The paper is attracted, showing the presence of static charge.
Quantitative tip (optional): Using a simple electroscope, the angular deflection of the leaves can be related to the charge transferred, giving a rough estimate of the magnitude of the charge produced by friction.
1.3 Detecting electrostatic charge (AO2)
- Electroscope: a metal rod with thin aluminium leaves. When a charged object is brought near, the leaves diverge because the like charges on the leaves repel each other.
- Charged‑rod‑paper‑leaf demonstration: a charged rod attracts lightweight paper pieces.
Safety note: static discharges can be unpleasant; avoid touching metal parts with wet hands and keep the demonstration away from sensitive electronic equipment.
What to look for (check‑list):
- Leaf divergence (indicates presence of charge).
- Direction of leaf motion when the charged object is moved closer (leaves move away – repulsion – or together – attraction).
- The electroscope does not reveal the sign of the charge; a separate test (e.g., bringing a known positively charged rod close) is needed to determine polarity.
1.4 Definition of electric field (AO1)
An electric field \(\mathbf{E}\) is a region of space in which a test charge \(q\) experiences a force \(\mathbf{F}\) given by
\(\displaystyle \mathbf{F}=q\,\mathbf{E}\)
- \(\mathbf{E}\) is a vector quantity – it has magnitude and direction.
- Direction of \(\mathbf{E}\): the direction of the force on a positive test charge.
- SI unit: newton per coulomb (N C⁻¹) or equivalently volt per metre (V m⁻¹).
- In vacuum \(k = \dfrac{1}{4\pi\varepsilon{0}} = 9.0\times10^{9}\ \text{N·m}^{2}\text{/C}^{2}\) where \(\varepsilon{0}=8.85\times10^{-12}\ \text{F·m}^{-1}\).
2 Simple Electric‑Field Patterns (AO1 & AO2)
2.1 Around a point charge
- Magnitude: \(E = k\,\dfrac{|q|}{r^{2}}\) (\(k = 1/4\pi\varepsilon_{0}\)).
- Why \(1/r^{2}\) ? The field lines spread out uniformly over the surface of an imaginary sphere of radius \(r\). The sphere’s area is \(4\pi r^{2}\); therefore the same total flux is distributed over a larger area as \(r\) increases, giving the inverse‑square dependence.
- Direction: away from the charge if \(q>0\); toward the charge if \(q<0\).
- Field‑line pattern: radial, equally spaced in angular direction, denser close to the charge.
2.2 Around a charged conducting sphere
- A conducting sphere in electrostatic equilibrium carries all excess charge on its outer surface.
- Outside the sphere (r ≥ R): the field is identical to that of a point charge \(Q\) placed at the centre:
\[
E = k\,\dfrac{|Q|}{r^{2}}
\]
- Inside the conducting material (r < R): the electric field is zero because the charges rearrange themselves until electrostatic equilibrium is reached.
- Direction: same as for a point charge of magnitude \(Q\) – outward if \(Q>0\), inward if \(Q<0\).
- Special note: the “zero field inside” statement is true only for a solid conductor in electrostatic equilibrium; a hollow cavity with no charge also has zero field (Gauss’s law).
2.3 Between two oppositely charged parallel conducting plates (edge effects ignored)
- When the plates are large compared with their separation, the field between them is essentially uniform.
- Magnitude: \[
E = \frac{\sigma}{\varepsilon_{0}}
\qquad\left(\sigma = \text{surface charge density on each plate (C m}^{-2}\right)
\]
- Independence of separation: as long as edge effects are ignored, the field does not depend on the distance between the plates.
- Direction: from the positively charged plate toward the negatively charged plate (the direction a positive test charge would move).
- Field‑line pattern: straight, parallel, equally spaced lines perpendicular to the plate surfaces.
3 Summary Table
| Configuration | Field magnitude | Direction of field lines | Key notes |
|---|---|---|---|
| Point charge \(q\) | \(E = k\displaystyle\frac{|q|}{r^{2}}\) | Away from \(q\) if \(q>0\); toward \(q\) if \(q<0\) | Radial; follows inverse‑square law |
| Charged conducting sphere (outside, \(r\ge R\)) | \(E = k\displaystyle\frac{|Q|}{r^{2}}\) | Same as point charge of magnitude \(Q\) | Inside conductor \(E=0\); excess charge resides on surface |
| Parallel plates (edge effects ignored) | \(E = \dfrac{\sigma}{\varepsilon_{0}}\) | From positive plate to negative plate | Uniform, independent of plate separation; straight parallel lines |
4 Key Points for Examination (AO1 & AO2)
- Define electric charge, give its unit (C), state the elementary charge \(e\) and note that charge is quantised and conserved.
- Explain how friction transfers electrons, give the glass‑rod‑silk (or plastic‑wool) experiment, and mention that only electrons move.
- Describe how an electroscope or a charged‑rod‑paper‑leaf set‑up detects static charge; include safety advice and the checklist of observable effects.
- State the formal definition of an electric field, the vector nature of \(\mathbf{E}\), its units (N C⁻¹ or V m⁻¹), and the relation \(\mathbf{F}=q\mathbf{E}\).
- For each configuration (point charge, conducting sphere, parallel plates):
- write the correct expression for the magnitude of \(E\) (including the constant \(k = 1/4\pi\varepsilon_{0}\) where appropriate),
- state the direction of the field lines based on the sign of the charge(s),
- mention any special conditions (zero field inside a conductor in electrostatic equilibrium, uniform field between plates, independence of plate separation, inverse‑square dependence for a point charge).
- Remember that edge (fringe) effects are ignored unless the question explicitly asks for them.