Induced Electric Fields and Their Radial Profile Around a Solenoid

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Induced electric field

An electric field produced by changing magnetic flux; it can exert forces on charges and exist without a conducting path.

Faraday’s law

The circulation of the electric field around a closed path equals the negative rate of change of magnetic flux: $\oint \vec{E} \cdot d\vec{l}=-d\Phi_m/dt$.

Induced versus electrostatic fields

An induced field can have nonzero circulation around a closed path; an electrostatic field has zero circulation.

Induced field with circular symmetry

For a circular path of radius $r$, its magnitude is $E=|\varepsilon|/(2\pi r)$.

Induced field around a long solenoid

At fixed time, its magnitude grows with radius inside the solenoid and falls as $1/r$ outside.

Quiz questions

1. When changing magnetic flux induces an emf in a circuit without a battery, what supplies work to the charges?Answer: The induced electric field produced by the changing flux.
2. How is induced emf related to the electric field around a closed circuit?Answer: $\varepsilon = \oint \vec{E} \cdot d\vec{l}$
3. A circular path of radius $0.50\,\text{m}$ has an induced emf of $6.0\,\text{V}$. What is the approximate field magnitude along the path?Answer: $1.9\,\text{V/m}$
4. If a conducting coil is removed while the magnetic flux through its location continues to change, what happens to the induced electric field?Answer: It can still exist in free space, though no wire carries current.
5. What is the magnetic-field magnitude inside an ideal long solenoid with $n$ turns per unit length and current $I$?Answer: $B=\mu_0 nI$