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Electric Fields

Electric Fields. An electric field is a vector field that describes the influence exerted by electric charges on other charges, whereby a test charge experiences a force F = q E. The field originates from static charges according to Coulomb’s law: E = (1/4πϵ₀)(q/r²) ṙ̂, where ϵ₀ is the vacuum permittivity and r is the distance from the source charge. For continuous charge distributions the superposition principle applies, summing contributions from each infinitesimal element. The field lines point away from positive and toward negative charges, with density proportional to local field strength.

Theoretical Context

In classical electromagnetism, Maxwell’s equations govern electric fields: Gauss’s law (∇·E = ρ/ϵ₀) links divergence of E to charge density ρ, Faraday’s law (∇×E = –∂B/∂t) expresses how temporal changes in magnetic field B induce electric fields, and the absence of magnetic charges leads to ∇·B = 0. The continuity equation ∂ρ/∂t + ∇·J = 0 ensures charge conservation, linking current density J to time-varying fields. In a vacuum or linear isotropic media, the electric displacement field D is defined by D = ϵ₀E + P (with P the polarization), and the total electric field derives from both free and bound charges, shaping a wide array of static and dynamic phenomena in electrostatics, dielectrics, and electrodynamics.

Electric Potential and Field Relationship

The electric field and electric potential V are related by E = −∇V (the field is the negative gradient of the potential). In one dimension: E = −dV/dx. This means field lines point in the direction of decreasing potential. For a positive test charge, work done moving against the field (to higher potential) requires positive external work: W = qΔV.

For a point charge Q: V = kQ/r and E = kQ/r² (pointing outward for positive Q). The potential decreases as 1/r while the field decreases as 1/r² — the potential is a scalar (easier to calculate for multiple charges by simple addition), while the field is a vector sum.

Uniform Electric Fields

A uniform electric field exists between parallel conducting plates with opposite charges (a capacitor). If the plates have charge density σ, the field between them is E = σ/ε₀, pointing from the positive to the negative plate. The potential difference (voltage) is V = Ed, where d is the plate separation. A charge q in this field experiences a constant force F = qE — like a gravitational field, but electrically driven.

Uniform fields are fundamental to cathode ray tubes, particle accelerators, and the measurement of the electron charge-to-mass ratio (Millikan oil drop experiment).

Gauss's Law and Field Calculation

For charge distributions with symmetry, Gauss's law ∮E·dA = Q_enc/ε₀ provides a direct route to the electric field. For an infinite uniformly charged plane with surface charge density σ: E = σ/(2ε₀) on each side. For a sphere with total charge Q: outside (r > R), E = kQ/r² as if all charge were at the centre; inside (r < R) for a conductor, E = 0.

Worked Example: Field at a Point

A positive charge Q = 2 μC is at the origin. Find the electric field at a point 0.3 m away.

E = kQ/r² = 8.99×10⁹ × 2×10⁻⁶ / 0.09 ≈ 2.0×10⁵ N/C, directed radially outward from the charge.

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Electric Dipoles and Polar Molecules

An electric dipole consists of two equal and opposite charges +q and −q separated by distance d. The dipole moment p = qd points from negative to positive. In a uniform electric field E, a dipole experiences a torque τ = p × E that aligns it with the field, and a potential energy U = −p·E. This is why polar molecules (like water, H₂O) align in electric fields and why microwave ovens heat food — the oscillating field rotates water molecules, generating heat through molecular friction.

In non-uniform fields, dipoles also experience a net force (gradient force) proportional to ∇(p·E). This force is exploited in optical tweezers, dielectrophoresis (manipulating cells and particles), and in the attraction of hair to a charged comb.

Key Equations Summary

Electric field definition: E = F/q — force per unit test charge. Point charge field: E = kQ/r². Uniform field: E = V/d between parallel plates. Energy stored: PE = qV (work done moving charge q through potential V). Gradient relation: E = −∇V (field points toward decreasing potential). Field line density represents field magnitude; lines never cross and originate on positive charges, terminating on negative charges or at infinity. The concept of field was introduced by Faraday to explain action-at-a-distance forces and underlies all of electromagnetic theory including Maxwell's equations and light.

References and further reading