Electricity and Magnetism · Problem 7.6

ALTERNATING-CURRENT CIRCUITS — Edward Purcell & David Morin

Physics — E&M · faradays-law-solenoid-electric-field

Problem

Growing current in a solenoid ** An infinite solenoid has radius RR and nn turns per unit length. The current grows linearly with time, according to I(t)=CtI(t) = Ct. Use the integral form of Faraday's law to find the electric field at radius rr, both inside and outside the solenoid. Then verify that your answers satisfy the differential form of the law.
Intuitive explanation

The key insight is that a changing magnetic flux through the solenoid induces a circulating electric field—not a radial one—by symmetry. Use Faraday's law with a circular loop centered on the solenoid axis to relate the induced E\mathcal{E} to the rate of flux change, then solve for E(r)E(r). The differential form ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} will confirm your result by showing both sides match when you compute the curl in cylindrical coordinates.

Intuitive explanation

The key insight is that a changing magnetic flux through the solenoid induces a circulating electric field—not a radial one—by symmetry. Use Faraday's law with a circular loop centered on the solenoid axis to relate the induced E\mathcal{E} to the rate of flux change, then solve for E(r)E(r). The differential form ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} will confirm your result by showing both sides match when you compute the curl in cylindrical coordinates.

Related theory

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