Problems on Induction: UNIZOR.COM - Physics4Teens - Electromagnetism - Laws of Induction
Channel: Zor Shekhtman on UNIZOR Education
Date: 02/07/2020

UNIZOR.COM - Physics through a Looking Glass of Mathematician

Notes to a video lecture on http://www.unizor.com

Problems on Electromagnetic Induction

Problem A

The following experiment is conducted in the space with Cartesian coordinates.
Two infinitely long parallel wires in XY-plane are parallel to X-axis, one at Y-coordinate y=a and another at Y-coordinate y=−a (assuming a is positive).
These two wires are connected by a third wire positioned along the Y-coordinate between points (0,a) and (0,−a).
The fourth wire, parallel to the third one, also connects the first two, but slides along the X-axis, always maintaining its parallel position to Y-axis. The X-coordinate of its position is monotonously increasing with time t according to some rule x=x(t).
All four wires are made of the same material and the same cross-section with the electrical resistance of a unit length r.

There is a uniform magnetic field of intensity B with field lines parallel to Z-axis.
Initial position of the fourth wire coincides with the third one, that is x(0)=0.

What should the function x(t) be to assure the generation of the same constant electric current I0 in the wire loop?
What is the speed of the fourth rod at initial time t=0?



Problem B

A rectangular wire frame in a space with Cartesian coordinates rotates with variable angular speed ω(t) in a uniform magnetic field B.
In the beginning at t=0 the wire frame is at rest, ω(0)=0. As the time passes, the angular speed is monotonously increasing from initial value of zero to some maximum. This models turning the rotation on.
The axis of rotation is Z-axis.
The initial position of a wire frame is that its plane coincides with XZ-plane.

The magnetic field lines are parallel to X-axis.
So the angle between the magnetic field lines and the wire frame plane φ(t) at t=0 equals to zero.
The sides of a wire frame parallel to Z-axis (those, that cross the magnetic field lines) have length a, the other two sides have length b.

Determine the generated electromotive force (EMF) in this wire frame as a function of time t.