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

__Problems on__

Electromagnetic Induction

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

*and another at Y-coordinate*

**y=a***(assuming*

**y=−a***is positive).*

**a**These two wires are connected by a third wire positioned along the Y-coordinate between points

*and*

**(0,a)***.*

**(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

*according to some rule*

**t***.*

**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

*with field lines parallel to Z-axis.*

**B**Initial position of the fourth wire coincides with the third one, that is

*.*

**x(0)=0**What should the function

*be to assure the generation of the same constant electric current*

**x(t)***in the wire loop?*

**I**_{0}What is the speed of the fourth rod at initial time

*?*

**t=0***Solution*

The area of a wire frame is changing with time:

*.*

**S(t)=2a·x(t)***Magnetic flux*through this wire frame is

*.*

**Φ(t)=B·S(t)**Therefore, the magnitude of the generated electromotive force or voltage

*, as a function of time, is*

**U(t)**

**U(t) =**d**Φ(t)/**d**t = 2B·a·x'(t)**The resistance

*of the wire loop, as a function of time, is a product of a resistance of a unit length of a wire*

**R(t)***by the total length of all four sides of a wire rectangle*

**r***.*

**L=4a+2x(t)***[*

**R(t) = r·L = 2r·***]*

**2a+x(t)**The electric current

*in a wire loop, according to the Ohm's Law, is*

**I(t)***{*

**I(t) = U(t)/R(t) =**

= 2B·a·x'(t) /= 2B·a·x'(t) /

*[*

**2r·***]}*

**2a+x(t)***{*

**=**

= B·a·x'(t) /= B·a·x'(t) /

*[*

**r·***]}*

**2a+x(t)**This current has to be constant and equal to

*. This leads us to a differential equation*

**I**_{0}

**I**_{0}=I(t)*{*

**I**_{0}= B·a·x'(t) /*[*

**r·***]}*

**2a+x(t)**Simplifying this equation, obtain

*[*

**I**_{0}·r / (B·a) = x'(t) /*]*

**2a+x(t)**[

*]*

**I**_{0}·r / (B·a)*[*

**·**d**t =**

=d=

*]*

**2a+x(t)***[*

**/***]*

**2a+x(t)**Integrating,

[

*]*

**I**_{0}·r / (B·a)

**·t + C = ln(2a+x(t))**

**2a + x(t) = C·e**^{I0·r·t / (B·a)}Since

*,*

**x(0)=0**

**C=2a***[*

**x(t) = 2a·***]*

**e**^{I0·r·t / (B·a)}− 1Speed of the motion of the fourth wire along the X-axis is

*[*

**x'(t) = 2a·e**^{I0·r·t/(B·a)}·*]*

**I**_{0}·r/(B·a)or

*[*

**x'(t) = e**^{I0·r·t/(B·a)}·*]*

**2I**_{0}·r/BAt time

*the initial speed is*

**t=0**

**x'(0) = 2I**_{0}·r / B*Problem B*

A rectangular wire frame in a space with Cartesian coordinates rotates with variable angular speed

*in a uniform magnetic field*

**ω(t)***.*

**B**In the beginning at

*the wire frame is at rest,*

**t=0***.*

**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

*at*

**φ(t)***equals to zero.*

**t=0**The sides of a wire frame parallel to Z-axis (those, that cross the magnetic field lines) have length

*, the other two sides have length*

**a***.*

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

*.*

**t**Solution

The angle

*between the magnetic field lines and the*

**φ(t)**wire frame plane is changing with time. In the initial position, when

the wire frame coincides with XZ-plane, this angle is 0°, since magnetic

field lines are stretched along the X-axis.

As the wire frame rotates with variable angular speed

*, the angle between magnetic field lines and the wire frame plane*

**ω(t)***and the angular speed*

**φ(t)***are related as follows*

**ω(t)***d*

**φ/**d**t = φ'(t) = ω(t)**This is sufficient to determine the value of

*by integration of the angular speed on a time interval [*

**φ(t)***]*

**0,t**

**φ(t) = ∫**d_{[0,t]}ω(τ)·**τ***Magnetic field flux*

*flowing through a wire frame depends on the intensity of the magnetic field*

**Φ(t)***, area of a wire frame*

**B***and angle*

**S=a·b***the plane of a wire frame makes with magnetic field lines.*

**φ(t)**

**Φ(t) = B·S·sin(φ(t))**The electromotive force (voltage)

*generated by rotating wire frame equals to a rate of change (first derivative) of the magnetic field flux*

**U(t)**

**U(t) =**d**Φ(t)/**d**t =**

= B·S·cos(φ(t))·φ'(t) =

= B·S·cos(φ(t))·ω(t)= B·S·cos(φ(t))·φ'(t) =

= B·S·cos(φ(t))·ω(t)

where angle

*can be obtained by an integration of an angular speed presented above.*

**φ(t)**