Induced Electric Fields The induced electric field \(12 \mathrm{cm}\) from the axis of a 10 -cm-radius solenoid is \(45 \mathrm{V} / \mathrm{m} .\) Find the rate of change of the solenoid's magnetic field.

Imagine you're holding a magnet near a coil of wire. If you suddenly move that magnet, an electric current seems to magically appear in the wire. This is the principle behind an induced electric field. As per Faraday's Law, when the magnetic environment of a coil changes, an electric field is induced in and around the coil.

This happens because changing magnetic fields create a sort of push on the electric charges in the wire, making them move. In the example from the textbook, a 12 cm distance from the axis of a solenoid is given an induced electric field value of 45 V/m. This value indicates the strength of the electric field that's been created by the changing magnetic field within the solenoid.

Magnetic flux is a measure of the number of magnetic field lines passing through a given area. To visualize magnetic flux, picture a net catching fish in a river. The number of fish caught in the net represents the magnetic flux, where the fish are the magnetic field lines, and the net is the area in question.

The unit for magnetic flux is the weber (Wb), and it is a product of the area through which the field lines pass and the magnetic field itself. It's a very useful concept when we deal with Faraday's Law since the induced electric field is related to the rate at which the magnetic flux changes over time.

This is all about how quickly the magnetic field is changing in strength or direction. Just like how a car's speed can change quickly or slowly, the magnetic field inside a solenoid can also change at different rates.

For our textbook exercise, the solenoid's magnetic field changing rate is what we're trying to find. When the question mentions a 'rate of change of the solenoid's magnetic field,' it's asking us to figure out how fast the magnetic field's strength is increasing or decreasing over time, measured in teslas per second (T/s).

A solenoid is essentially a coil of wire, usually wound tightly in the shape of a cylinder. When electric current flows through it, it creates a uniform magnetic field inside the solenoid, much like a bar magnet. It's an important piece of technology found in many places, from car starters to doorbells.

In our exercise, we encounter a solenoid that has its magnetic field changing. As the magnetic field changes, this solenoid becomes the site of an induced electric field. Its very form, with loops of wire, ensures this electric field is generated in a consistent manner due to its geometry aiding in the induction process.