Image source: https://commons.wikimedia.org/wiki/File:Electromagnetic_induction.svg

## Introduction

Hello it's a me again Drifter Programming! Today we continue with **Electromagnetism** to get into **Electromagnetic Induction Exercises**. We will get into separate examples around each article's topic to stay organized. I will try to get into the most important aspects, cause getting into a lot of exercises would be exhausting! Of course reading the previous articles will help you a lot, something that I suggest you to do...So, without further do, let's get started!

## Faraday's Law

Let's start out with maybe the most important law of Induction: "Faraday's law".

### Example 1:

Let's consider a plane circular loop of conducting wire with radius r = 10cm and N = 15 turns. This loop is placed in a uniform magnetic field and the direction of the magnetic field makes an angle of 30° to the normal vector/direction of the loop. The magnetic field strength B is being increased with a constant rate from B1 = 1T to B2 = 5T in a time interval of Δt = 10s. Calculate the emf generated around the loop if the electrical resistance is R = 15Ω, and also the (induced) current that flows as the magnetic field is being increased.

To calculate the magnetic flux (which is needed in Faraday's law) we first have to calculate the area of the loop and also the magnetic field perpendicular to the loop.

The area of the loop, which is a circle is:

The magnetic field component perpendicular to the loop is:

Therefore, the initial magnetic flux (for B1) is:

And the final flux (for B2) is:

Lastly, the Emf generated is:

From Ohm's law the current in response to the emf is:

Of course the emf and current actually have a negative sign...

### Example 2:

Consider an MRI technician moving his hand from a region of very low magnetic field strength (0T) into an MRI scanner that has a 2T magnetic field with his fingers pointing in the direction of the field. Find the average emf induced in his wedding ring, givien it has a diameter of 2.2cm and assuming it takes 0.25s to move it into the field. Lastly, also find out if the current would generate a significant amount of heat / change in temperature for a resistance of R = 1mΩ.

The area of the ring is:

Therefore, the magnetic flux for B = 2T is:

The average emf is calculated as:

The produced heat is:

which is not a significant amount of heat!

## Motional Emf

Let's now continue on with Motional Emf!

### Example 1:

Consider a space shuttle orbiting the earth while being tethered to a satellite. A 5kV emf is predicted to be induced in the 20km long tether while moving at orbital speed in the Earth's magnetic field. Knowing that the tether has a length of 20km, that the orbital speed is 7.8 Km/s and that the magnetic field of the Earth has a strength of 5x10^-5 T, calculate the motional emf of the space shuttle.

We can clearly see that this is a straightforward application of the motional emf equation:

By filling in the values with their corresponding units we get:

The value obtained is greater then the predicted 5kV voltage, since the orbital motion of the tether is not perpendicular to the Earth's field. 7.8kV is the maximum emf value that is obtained when θ = 90 and so sinθ = 1. Το get a value closer to 5kV we have to put the tether in an angle corresponding to the Earth's magnetic field!

### Example 2:

Image source: http://farside.ph.utexas.edu/teaching/302l/lectures/node88.html#s9.4

Let's now consider a moving rod of length l = 0.2m and with a velocity of v = 0.1m/s through a magnetic field-strength B = 0.1T directed "into the page". The resistance of this circuit is R = 0.002Ω.

Calculate:

- The emf generated around the circuit
- The current that flows around the circuit
- The magnitude and direction of the force acting on the moving rod due to the current flowing along it
- The rate at which work must be performed on the rod in order to keep it moving at constant velocity against this force
- The rate at which electrical energy is generated
- The rate at which energy is converted into heat due to the resistivity of the circuit

According to motional emf the emf generated is:

This emf acts in the anti-clockwise direction.

The anti-clockwise current driven by the motional emf, using Omh's law, is:

Since the rod carries a current I, which flows perpendicular to a magnetic field, the total magnetic force acting on the rod is:

Using the right-hand rule this force is parallel to the vector IxB that follows the force to the left and so opposes the motion producing the emf.

To keep the rod moving with a constant velocity, the following work needs to be performed:

The rate at which electric charges acquire energy in the circuit is

The rate at which electrical energy is converted into heat energy in the circuit is:

From all this we can see that the total mechanical work done eventually ends up as heat dissipated in the circuit...

## Lenz's Law

Lenz's Law tells us that the induced current produces a magnetic field that opposes the change in the magnetic flux. **Lenz's** law states that "*The induced current flows in such a direction so that it opposes the changes that induced it*".

### Application 1:

Let's consider a magnetic field that is directed "into the page" and a circular loop of wire parallel to the page and so perpendicular to the magnetic field.

What happens when the strength of the magnetic field:

- increases
- decreases
- remains the same

**1. Increases**

The magnetic field heading into the page is increasing and so a induced magnetic field heading out of the page is being produced. From the right-hand thumb rule we find that the induced current is flowing in the anti-clockwise direction.

**2. Decreases**

The magnetic field heading into the page is decreasing and so a induced magnetic field heading into the page is being produced. From the right-hand thumb rule we find that the induced current is flowing in the clockwise direction.

**3. Remains the same**

When the magnetic field is staying the same then there is no magnetic flux change and thus no electromagnetic induction. This means that no current is flowing through the wire, cause an induced emf and induced current only occurs when the magnetic flux is being changed at some rate.

### Application 2:

Image source is Reference 6!

Consider the north pole of a magnet heading towards a loop of wire (coil). If we are looking at the loop from above the magnet will the induced current be circulating clockwise or counterclockwise?

Magnetic field lines start from north pole and are heading into south poles and so to oppose the effect of the magnetic flux increase. the current would be flowing in the counterclockwise direction, which causes a magnetic field in the opposing direction that lowers the magnetic flux.

## Induced Electric Fields

### Circular Coil:

Let's now get into the induced electric field of the previous circular coil.

Because of cylindrical symmetry the electric field integral simplifies into:

Combined with Faraday's integral form we get:

### Solenoid:

Let's now consider a solenoid. What is the electric field induced when it's affected by a changing magnetic field?

The magnetic field confined to the interior has been proven to be:

The magnetic flux through a circular path whose radius *r* is greater than the solenoid radius R is:

Because of the system's cylindrical symmetry the induced field is tangent to that path and has a constant magnitude which means that:

For a radius r <R we get:

## Eddy Currents and Maxwell Equations

The complex mathematics of the last two would make even one example large and very difficult to explain, cause I am not a Physist nor a Mathematician, but just understand the Physics principles and theorems very easily...

If you really want to read about them you can check the following two links:

- https://www.sciencedirect.com/science/article/pii/S0168927413000779
- http://www.danfleisch.com/maxwell/chapter1/index.shtml

which seem to be very well explained!

### REFERENCES:

- http://farside.ph.utexas.edu/teaching/302l/lectures/node95.html
- https://pressbooks.bccampus.ca/collegephysics/chapter/faradays-law-of-induction-lenzs-law/
- https://opentextbc.ca/physicstestbook2/chapter/motional-emf/
- http://farside.ph.utexas.edu/teaching/302l/lectures/node97.html
- https://physics.info/lenz/problems.shtml
- https://phys.libretexts.org/TextBooks_and_TextMaps/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_II_-_Thermodynamics%2C_Electricity%2C_and_Magnetism_(OpenStax)/13%3A_Electromagnetic_Induction/13.E%3A_Electromagnetic_Induction_(Exercises)

Mathematical equations that I had in this post where drawn using quicklatex!

## Previous posts about Electromagnetism

### Electric fields:

Getting into Electromagnetism -> electromagnetim, electric charge, conductors, insulators, quantization

Coulomb's law with examples -> Coulomb's law, superposition principle, Coulomb constant, how to solve problems, examples

Electric fields and field lines -> Electric fields, Solving problems around Electric fields and field lines

Electric dipoles -> Electric dipole, torque, potential and field

Electric charge and field Exercises -> examples in electric charges and fields

### Electric flux:

Electric flux and Gauss's law -> Electric flux, Gauss's law

Applications of Gauss's law (part 1) -> applying Gauss's law, Gauss applications

Applications of Gauss's law (part 2) -> more Gauss applications

Electric flux exercises -> examples in electric flux and Gauss's law

### Electric potential:

Electric potential energy -> explanation of work-energy, electric potential energy

Calculating electric potentials -> more stuff about potential energy, potential, calculating potentials

Equipotential surfaces and potential gradient -> Equipotential surface, potential gradient

Millikan's Oil Drop Experiment -> Millikan's experiment, electronvolt

Cathode ray tubes explained using electric potential -> cathode ray tube explanation

Electric potential exercises (part 1) -> applications of potential

Electric potential exercises (part 2) -> applications of potential gradient, advanced examples

### Capacitance:

Capacitors (Condensers) and Capacitance -> Capacitors, capacitance, calculating capacitance

How to solve problems around Capacitors -> combination, solving problems, simple example

Electric field energy and density -> Electric field energy, energy density

Dielectric materials -> Dielectrics, dielectric constant, permittivity and strength, how to solve problems

Electric capacitance exercises -> examples in capacitance, energy density and dielectrics

### Current, resistance and EMF:

Electric current -> Electric current, current density

Electrical resistivity and conductivity -> Electrical resistivity, conductivity, thermal coefficient of resistivity, hyperconductivity

Electric resistance -> Resistance, temperature, resistors

Electromotive Force (EMF) and Internal resistance -> Electromotive force, internal resistance

Power and Wattage of Electronic Circuits -> Power in general, power/wattage of electronic circuits

Electric current, resistance and emf exercises -> exampes in all those topics

### Direct current (DC) circuits:

Resistor Combinations -> Resistor combinations, how to solve problems

Kirchhoff's laws with applications -> Kirchhoff's laws, how to solve problems, applications

Electrical measuring instruments -> what are they?, types list, getting into some of them, an application

Electronic circuits with resistors and capacitors (R-C) -> R-C Circuit, charging, time constant, discharging, how to apply

RC circuit exercises -> examples in Kirchhoff, charging, discharging capacitor with/without internal resistance

### Magnetic field and forces:

Magnetic fields -> Magnetism, Magnetic field

Magnetic field lines and Gauss's law of Magnetism -> magnetic field lines, mono- and dipoles, Flux, Gauss's law of magnetism

The motion of charged particles inside of a magnetic field -> straight-line, spiral and helical particle motion

Applications of charged particle motion -> CERN, Cyclotrons, Synchrotrons, Cavity Magetron, Mass Spectrometry and Magnetic lens

Magnetic force applied on Current-Carrying Conductors -> magnetic force on current-carrying conductors/wires, proofs

Magnetic force and torque applied on current loops (circuits) -> magnetic force on current loops, magnetic moment and torque

Explaining the Physics behind Electromotors -> tesla, history and explaining the physics behind them

Magnetic field exercises -> examples in magnetic force, magnetic flux, particle motion and forces/torque on current-carrying conductors

### Magnetic field sources:

Magnetic field of a moving charged particle -> moving charge, magnetic field, force between parallel charged particles

Magnetic field of current-carrying conductors -> magnetic field of current, Biot-Savart law

Force between parallel conductors and the magnetic field of a current loop-> force between parallel conductors, magnetic field of current loop

Ampere's law and Applications -> Ampere's law, applications

Magnetic materials -> Magnetic materials, classification and types, material examples

Displacement current -> Displacement current, Extension of Ampere's law

Exercises in Magnetic field sources -> examples all around magnetic field sources

### Electromagnetic Induction:

Electromagnetic Induction and Faraday's law -> Electromagnetic Induction, Experiments, Faraday's law

Motional Electromotive Force (Emf) -> Motional Emf, Faraday's law and motional emf, generalization

Lenz's law and Induced Electric fields -> Lenz's law, Induced Electric Fields

Eddy Currents and Applications -> Εddy currents, applications (brakes, testing, others)

Maxwell's equations -> What they are, each equation analyzed separately

And this is actually it for today's post! Next time in Physics we will get into Mutual Induction, which gets us to the new "chapter" of "Mutual and Self Induction" and has a lot of interesting things to offer!

C ya!