Electric Dipole and a Dipole in an External Field
A water molecule, a bit of paper near a charged comb, the antenna of a radio – all behave like a pair of equal and opposite charges a small distance apart: an electric dipole. On this page you derive its field on the axis and on the equatorial line (two favourite board-exam derivations), solve Example 1.10, and twist a dipole in a field to see the torque τ = pE sin θ.
- Field of a point charge E = (1/4πε0) q/r²; fields add as vectors (superposition).
- Algebra: (r + a)² − (r − a)² = 4ar; if r ≫ a, then r² − a² ≈ r² and r² + a² ≈ r².
- Torque of a couple = one force × perpendicular distance between the two forces; cross product |A × B| = AB sin θ (pages 1 and 3).
1. What is an electric dipole?
An electric dipole (വൈദ്യുത ദ്വിധ്രുവം) is a pair of equal and opposite point charges q and −q separated by a distance 2a. The line joining them is the axis; by convention the direction from −q to q is the direction of the dipole. The mid-point is the centre of the dipole.
The total charge is zero, but the field is not zero: the two charges are at different places, so their fields do not cancel exactly. Far away (r ≫ 2a) they nearly cancel, so the dipole field falls off faster than 1/r². Let us calculate it.
2. Field on the axis of a dipole (Fig 1.20 a)
E = q4πε0 4ar(r2 − a2)2 p̂ (1.14) r ≫ a: E = 2p4πε0r3 (1.20) – along p
3. Field on the equatorial line (Fig 1.20 b)
E = − 2qa4πε0(r2 + a2)3/2 p̂ (1.17) r ≫ a: E = − p4πε0r3 (1.21) – opposite to p
4. Dipole moment and the 1/r³ law
Equations (1.15) and (1.18) contain q and a only as the product qa. This suggests the dipole moment (ദ്വിധ്രുവ മൊമെന്റ്):
p = q × 2a p̂ (1.19) – magnitude q × 2a, direction from −q to q; SI unit C m (coulomb metre)
- Far away the dipole field falls off as 1/r³ (a single charge: 1/r²).
- At the same distance, the axial field is twice the equatorial field: 2p/(4πε0r³) against p/(4πε0r³); axial is along p, equatorial is opposite to p.
- In general the field depends on r and on the angle between r and p.
- Point dipole: let 2a → 0 and q → ∞ with p = q × 2a fixed. For a point dipole, (1.20) and (1.21) are exact at every r.
Charges ±1 nC; 1 grid square = 10 cm. Drag the probe ⊕. On the axis or the equatorial line the readout compares the exact superposition result with the far-field formulas 2p/(4πε0r³) and p/(4πε0r³) – move far away and watch them agree.
5. Physical significance – polar and non-polar molecules
In most molecules the centre of positive charge and the centre of negative charge coincide, so the dipole moment is zero – CO2 and CH4 are such non-polar molecules (അധ്രുവീയ തന്മാത്ര). They develop a dipole moment only when an electric field is applied (an induced dipole). In some molecules the centres do not coincide, so they have a permanent dipole moment even without a field: polar molecules (ധ്രുവീയ തന്മാത്ര) – water, H2O, is the classic example.
Water: the O atom pulls the shared electrons towards itself, so O carries a small negative charge (δ−) and each H a small positive charge (δ+); the bent shape stops the two bond dipoles from cancelling. CO2 is straight and symmetric: its two bond dipoles cancel. In a field, the electron cloud of any molecule shifts slightly against E – an induced dipole along E. (Simplified sketch, not to scale.)
A microwave oven fills the food with an electric field that reverses direction billions of times a second. The polar water molecules keep trying to turn and line up with the field (exactly the torque of section 7) and jostle their neighbours – the food heats up. Dry plates without water stay much cooler.
6. Example 1.10 – field of a ±10 μC dipole (Fig 1.21)
Example 1.10. Two charges ±10 μC are placed 5.0 mm apart. Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in Fig 1.21(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in Fig 1.21(b).
a = 2.5 mm = 0.25 cm, r = 15 cm. (a) Find the field of each charge at P (distances r − a and r + a) and subtract. (b) Each charge is √(r² + a²) from Q; only the components parallel to the axis survive: E = 2 E1 × a/√(r² + a²). Then compare with the short formulas 2p/(4πε0r³) and p/(4πε0r³), with p = q × 2a.
7. Dipole in a uniform external field (Fig 1.22)
Place a permanent dipole p in a uniform electric field (ഏകീകൃത വൈദ്യുത മണ്ഡലം) E, making an angle θ with it. The charge q feels a force qE; the charge −q feels −qE. These are equal and opposite, so the net force is zero. But they act at different points – they form a couple, which produces a torque (ടോർക്ക് (ബലാഘൂർണം)).
τ = pE sin θ τ = p × E (1.22)
The torque tries to align p with E. θ = 0°: τ = 0, stable equilibrium (സന്തുലനം (സ്ഥിര/അസ്ഥിര)); θ = 90°: maximum torque pE; θ = 180°: τ = 0 again, but unstable – the slightest nudge turns the dipole right round. Unit of torque: N m.
Set the starting angle, then release. With a little friction (damping) the dipole swings about θ = 0 and settles along E. Try θ = 180°: the torque is zero, but release from 175° and it swings all the way round – θ = 180° is unstable.
8. Dipole in a non-uniform field (Fig 1.23) – the comb and the paper
If the field is not uniform, the forces on +q and −q are no longer equal, so there is a net force as well (and in general a torque). Take the simple cases p parallel or antiparallel to E (torque zero):
- p parallel to E: net force in the direction of increasing field.
- p antiparallel to E: net force in the direction of decreasing field.
A comb run through dry hair becomes charged, but the paper is neutral. The comb's field polarises the paper – it induces a dipole moment along the field (p parallel to E). The comb's field is non-uniform: it is stronger closer to the comb. A dipole with p ∥ E is pulled towards the stronger field, so the paper moves towards the comb – whatever the sign of the comb's charge.
The same effect makes a rubbed balloon stick to a wall, makes dust cling to a TV screen or a ceiling fan blade, and is used in electrostatic precipitators, where dust particles in chimney smoke are charged or polarised and pulled out by strong non-uniform fields.
- 📄 Electric dipole moment Wikipedia
Dipoles, polar molecules and torque in a field.
9. Check yourself
- Dipole: ±q separated by 2a; p = q × 2a, from −q to +q; unit C m; total charge zero but field not zero.
- Axial line: E = (q/4πε0)·4ar/(r² − a²)² → 2p/(4πε0r³) along p. Equatorial line: E = 2qa/[4πε0(r² + a²)3/2] → p/(4πε0r³) opposite to p. Dipole field ∝ 1/r³.
- Polar molecules (H2O) have permanent dipoles; non-polar ones (CO2, CH4) get induced dipoles in a field.
- Ex 1.10: axial 2.7 × 105 N/C along BP (short formula 2.66 × 105); equatorial 1.33 × 105 N/C along BA.
- Uniform field: net force 0, torque τ = p × E, τ = pE sin θ; aligns p with E (θ = 0 stable, 180° unstable).
- Non-uniform field: net force too – towards stronger field if p ∥ E (comb attracts paper).