10.4 Induced Electric Fields
LEARNING OBJECTIVES
- Connect the relationship between an induced emf from Faraday’s law to an electric field, thereby showing that a changing magnetic flux creates an electric field
- Solve for the electric field based on a changing magnetic flux in time
The fact that emfs are induced in circuits implies that work is being done on the conduction electrons in the wires. What can possibly be the source of this work? We know that it’s neither a battery nor a magnetic field, for a battery does not have to be present in a circuit where current is induced, and magnetic fields never do work on moving charges. The answer is that the source of the work is an electric field that is induced in the wires. The work done by in moving a unit charge completely around a circuit is the induced emf ; that is,
where represents the line integral around the circuit. Faraday’s law can be written in terms of the induced electric field as
There is an important distinction between the electric field induced by a changing magnetic field and the electrostatic field produced by a fixed charge distribution. Specifically, the induced electric field is nonconservative because it does net work in moving a charge over a closed path, whereas the electrostatic field is conservative and does no net work over a closed path. Hence, electric potential can be associated with the electrostatic field, but not with the induced field. The following equations represent the distinction between the two types of electric field:
(10.4.3)
Our results can be summarized by combining these equations:
EXAMPLE 10.4.1
Induced Electric Field in a Circular Coil
What is the induced electric field in the circular coil of Example 10.2.1 (and ??) at the three times indicated?
Strategy
Using cylindrical symmetry, the electric field integral simplifies into the electric field times the circumference of a circle. Since we already know the induced emf, we can connect these two expressions by Faraday’s law to solve for the induced electric field.
Solution
The induced electric field in the coil is constant in magnitude over the cylindrical surface, similar to how Ampere’s law problems with cylinders are solved. Since is tangent to the coil,
When combined with Equation 10.4.4, this gives
The direction of is counterclockwise, and circulates in the same direction around the coil. The values of are
Significance
When the magnetic flux through a circuit changes, a nonconservative electric field is induced, which drives current through the circuit. But what happens if in free space where there isn’t a conducting path? The answer is that this case can be treated as if a conducting path were present; that is, nonconservative electric fields are induced wherever , whether or not there is a conducting path present.
These nonconservative electric fields always satisfy Equation 10.4.4. For example, if the circular coil of Equation 10.2.3 were removed, an electric field in free space at would still