A long, thin solenoid has 900 turns per meter and radius 2.50 cm . The current in the solenoid

A long, thin solenoid has 900 turns per meter and radius...

A long, thin solenoid has 900 turns per meter and radius $2.50 \mathrm{~cm} .$ The current in the solenoid is increasing at a uniform rate of $36.0 \mathrm{~A} / \mathrm{s}$. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) $0.500 \mathrm{~cm}$ from the axis of the solenoid; (b) $1.00 \mathrm{~cm}$ from the axis of the solenoid?

Key Concepts

Symmetry in Electromagnetic Induction

Symmetry plays a critical role in simplifying the analysis of electromagnetic induction problems. When the geometry of the system, such as a long solenoid, exhibits symmetry, it allows the induced electric field to be assumed to have a consistent magnitude along chosen paths (like circles concentric with the solenoid axis). This property simplifies the application of Faraday's law and the integration needed to calculate the magnitude of the induced electric field at various radial distances.

Induced Electric Field in a Time-Varying Magnetic Field

A time-varying magnetic field produces an induced electric field, even in regions where there is no conductive path for conventional current flow. This induced electric field is non-conservative, meaning it cannot be described by a scalar potential globally, and its magnitude and direction can be obtained using the integral form of Faraday's law, often by considering circular loops centered on the axis of symmetry.

Magnetic Field in a Solenoid

The magnetic field inside an ideal solenoid is uniform and can be calculated based on the number of turns per unit length and the current passing through the solenoid. This concept is important because it provides the basis for determining the magnetic flux that links a loop or area, especially when the solenoid's current is changing with time.

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle that relates a time-varying magnetic flux through a closed loop to an induced electromotive force (EMF) in that loop. It mathematically expresses that the induced EMF is equal to the negative rate of change of magnetic flux, which is a central idea in explaining how changing magnetic fields give rise to electric fields in various physical systems.

Transcript

00:01 We're told that we have a long thin solenoid with 900 turns per meter, so that's in.

00:05 Of course, the permeativity here, mu not is 1 .25 times 10 to the minus 6 nune per ampiers.

00:11 And we increased the solenoid at a uniform rate.

00:14 D -i -d -t then is 36 ampers per second.

00:18 And the radius, which i call r, is 0 .025 meters.

00:24 Okay, then to find the electric field at a point zero to five meters.

00:30 Point near the center.

00:31 First, we're going to use e .d .s.

00:38 Is equal to minus the integral of dbdt .t.

00:49 Okay, well, the magnetic field, b is equal to mu not times the number of turns times the current i.

01:03 Therefore d b d t is equal to mu not times the number of turns times di d t and we know all of those values so then going back up to the integral the left side of the integral is the electric field times two pi r where the right side is equal to mu not times the number of turns times d i d t times pi r squared in general for the first case if we're trying to solve for the center so we'll start another page here so therefore the electric field is equal to mu not times the number of turns times d i d t multiplied by r the pies cancel out divided by two so plugging all those values of the expression we find that near the center the electric field is equal to 5 .09 times 10 to the minus 4, and then the units here would be newton's per coolum.

02:47 So we can go ahead and box that in...

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